Home | Metamath
Proof Explorer Theorem List (p. 210 of 449) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-28623) |
Hilbert Space Explorer
(28624-30146) |
Users' Mathboxes
(30147-44804) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lindfmm 20901 | Linear independence of a family is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) ⇒ ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑆 ↔ (𝐺 ∘ 𝐹) LIndF 𝑇)) | ||
Theorem | lindsmm 20902 | Linear independence of a set is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) ⇒ ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹 ⊆ 𝐵) → (𝐹 ∈ (LIndS‘𝑆) ↔ (𝐺 “ 𝐹) ∈ (LIndS‘𝑇))) | ||
Theorem | lindsmm2 20903 | The monomorphic image of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) ⇒ ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹 ∈ (LIndS‘𝑆)) → (𝐺 “ 𝐹) ∈ (LIndS‘𝑇)) | ||
Theorem | lsslindf 20904 | Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑈 = (LSubSp‘𝑊) & ⊢ 𝑋 = (𝑊 ↾s 𝑆) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊)) | ||
Theorem | lsslinds 20905 | Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝑈 = (LSubSp‘𝑊) & ⊢ 𝑋 = (𝑊 ↾s 𝑆) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ∈ (LIndS‘𝑋) ↔ 𝐹 ∈ (LIndS‘𝑊))) | ||
Theorem | islbs4 20906 | A basis is an independent spanning set. This could have been used as alternative definition of a basis: LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ (((LSpan‘𝑤) ‘𝑏) = (Base‘𝑤) ∧ 𝑏 ∈ (LIndS‘𝑤))}). (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) ⇒ ⊢ (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵)) | ||
Theorem | lbslinds 20907 | A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐽 = (LBasis‘𝑊) ⇒ ⊢ 𝐽 ⊆ (LIndS‘𝑊) | ||
Theorem | islinds3 20908 | A subset is linearly independent iff it is a basis of its span. (Contributed by Stefan O'Rear, 25-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) & ⊢ 𝑋 = (𝑊 ↾s (𝐾‘𝑌)) & ⊢ 𝐽 = (LBasis‘𝑋) ⇒ ⊢ (𝑊 ∈ LMod → (𝑌 ∈ (LIndS‘𝑊) ↔ 𝑌 ∈ 𝐽)) | ||
Theorem | islinds4 20909* | A set is independent in a vector space iff it is a subset of some basis. (AC equivalent) (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐽 = (LBasis‘𝑊) ⇒ ⊢ (𝑊 ∈ LVec → (𝑌 ∈ (LIndS‘𝑊) ↔ ∃𝑏 ∈ 𝐽 𝑌 ⊆ 𝑏)) | ||
Theorem | lmimlbs 20910 | The isomorphic image of a basis is a basis. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐽 = (LBasis‘𝑆) & ⊢ 𝐾 = (LBasis‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ 𝐵) ∈ 𝐾) | ||
Theorem | lmiclbs 20911 | Having a basis is an isomorphism invariant. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐽 = (LBasis‘𝑆) & ⊢ 𝐾 = (LBasis‘𝑇) ⇒ ⊢ (𝑆 ≃𝑚 𝑇 → (𝐽 ≠ ∅ → 𝐾 ≠ ∅)) | ||
Theorem | islindf4 20912* | A family is independent iff it has no nontrivial representations of zero. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑌 = (0g‘𝑅) & ⊢ 𝐿 = (Base‘(𝑅 freeLMod 𝐼)) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘f · 𝐹)) = 0 → 𝑥 = (𝐼 × {𝑌})))) | ||
Theorem | islindf5 20913* | A family is independent iff the linear combinations homomorphism is injective. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ · = ( ·𝑠 ‘𝑇) & ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘f · 𝐴))) & ⊢ (𝜑 → 𝑇 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) & ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) ⇒ ⊢ (𝜑 → (𝐴 LIndF 𝑇 ↔ 𝐸:𝐵–1-1→𝐶)) | ||
Theorem | indlcim 20914* | An independent, spanning family extends to an isomorphism from a free module. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ · = ( ·𝑠 ‘𝑇) & ⊢ 𝑁 = (LSpan‘𝑇) & ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘f · 𝐴))) & ⊢ (𝜑 → 𝑇 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) & ⊢ (𝜑 → 𝐴:𝐼–onto→𝐽) & ⊢ (𝜑 → 𝐴 LIndF 𝑇) & ⊢ (𝜑 → (𝑁‘𝐽) = 𝐶) ⇒ ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMIso 𝑇)) | ||
Theorem | lbslcic 20915 | A module with a basis is isomorphic to a free module with the same cardinality. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝑊 ≃𝑚 (𝐹 freeLMod 𝐼)) | ||
Theorem | lmisfree 20916* | A module has a basis iff it is isomorphic to a free module. In settings where isomorphic objects are not distinguished, it is common to define "free module" as any module with a basis; thus for instance lbsex 19868 might be described as "every vector space is free". (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → (𝐽 ≠ ∅ ↔ ∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘))) | ||
Theorem | lvecisfrlm 20917* | Every vector space is isomorphic to a free module. (Contributed by AV, 7-Mar-2019.) |
⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ LVec → ∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘)) | ||
Theorem | lmimco 20918 | The composition of two isomorphisms of modules is an isomorphism of modules. (Contributed by AV, 10-Mar-2019.) |
⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐺 ∈ (𝑅 LMIso 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 LMIso 𝑇)) | ||
Theorem | lmictra 20919 | Module isomorphism is transitive. (Contributed by AV, 10-Mar-2019.) |
⊢ ((𝑅 ≃𝑚 𝑆 ∧ 𝑆 ≃𝑚 𝑇) → 𝑅 ≃𝑚 𝑇) | ||
Theorem | uvcf1o 20920 | In a nonzero ring, the mapping of the index set of a free module onto the unit vectors of the free module is a 1-1 onto function. (Contributed by AV, 10-Mar-2019.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) ⇒ ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼–1-1-onto→ran 𝑈) | ||
Theorem | uvcendim 20921 | In a nonzero ring, the number of unit vectors of a free module corresponds to the dimension of the free module. (Contributed by AV, 10-Mar-2019.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) ⇒ ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) → 𝐼 ≈ ran 𝑈) | ||
Theorem | frlmisfrlm 20922 | A free module is isomorphic to a free module over the same (nonzero) ring, with the same cardinality. (Contributed by AV, 10-Mar-2019.) |
⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → (𝑅 freeLMod 𝐼) ≃𝑚 (𝑅 freeLMod 𝐽)) | ||
Theorem | frlmiscvec 20923 | Every free module is isomorphic to the free module of "column vectors" of the same dimension over the same (nonzero) ring. (Contributed by AV, 10-Mar-2019.) |
⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌) → (𝑅 freeLMod 𝐼) ≃𝑚 (𝑅 freeLMod (𝐼 × {∅}))) | ||
According to Wikipedia ("Matrix (mathemetics)", 02-Apr-2019, https://en.wikipedia.org/wiki/Matrix_(mathematics)) "A matrix is a rectangular array of numbers or other mathematical objects for which operations such as addition and multiplication are defined. Most commonly, a matrix over a field F is a rectangular array of scalars each of which is a member of F. The numbers, symbols or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.", and in the definition of [Lang] p. 503 "By an m x n matrix in [a commutative ring] R one means a doubly indexed family of elements of R, (aij), (i= 1,..., m and j = 1,... n) ... We call the elements aij the coefficients or components of the matrix. A 1 x n matrix is called a row vector (of dimension, or size, n) and a m x 1 matrix is called a column vector (of dimension, or size, m). In general, we say that (m,n) is the size of the matrix, ...". In contrast to these definitions, we denote any free module over a (not necessarily commutative) ring (in the meaning of df-frlm 20821) with a Cartesian product as index set as "matrix". The two sets of the Cartesian product even need neither to be ordered or a range of (nonnegative/positive) integers nor finite. By this, the addition and scalar multiplication for matrices correspond to the addition (see frlmplusgval 20838) and scalar multiplication (see frlmvscafval 20840) for free modules. Actually, there isn't a definition for (arbitrary) matrices: Even the (general) matrix multiplication can be defined using functions from Cartesian products into a ring (which are elements of the base set of free modules), see df-mamu 20925. By this, a statement like "Then the set of m x n matrices in R is a module (i.e., an R-module)" as in [Lang] p. 504 follows immediately from frlmlmod 20823. However, for square matrices there is the definition df-mat 20947, defining the algebras of square matrices (of the same size over the same ring), extending the structure of the corresponding free module by the matrix multiplication as ring multiplication. A "usual" matrix (aij), (i= 1,..., m and j = 1,... n) would be represented as element of (the base set of) (𝑅 freeLMod ((1...𝑚) × (1...𝑛))), and a square matrix (aij), (i= 1,..., n and j = 1,... n) would be represented as element of (the base set of) ((1...𝑛) Mat 𝑅). Finally, it should be mentioned that our definitions of matrices include the zero-dimensional cases, which is excluded in the definition of many authors, e.g. in [Lang] p. 503. It is shown in mat0dimbas0 21005 that the empty set is the sole zero-dimensional matrix (also called "empty matrix", see Wikipedia https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrices). 21005 The determinant is also defined for such an empty matrix, see mdet0pr 21131. | ||
This section is about the multiplication of m x n matrices. | ||
Syntax | cmmul 20924 | Syntax for the matrix multiplication operator. |
class maMul | ||
Definition | df-mamu 20925* | The operator which multiplies an m x n matrix with an n x p matrix, see also the definition in [Lang] p. 504. Note that it is not generally possible to recover the dimensions from the matrix, since all n x 0 and all 0 x n matrices are represented by the empty set. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
⊢ maMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ ⦋(1st ‘(1st ‘𝑜)) / 𝑚⦌⦋(2nd ‘(1st ‘𝑜)) / 𝑛⦌⦋(2nd ‘𝑜) / 𝑝⦌(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m (𝑛 × 𝑝)) ↦ (𝑖 ∈ 𝑚, 𝑘 ∈ 𝑝 ↦ (𝑟 Σg (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑗𝑦𝑘))))))) | ||
Theorem | mamufval 20926* | Functional value of the matrix multiplication operator. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ Fin) ⇒ ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐵 ↑m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑m (𝑁 × 𝑃)) ↦ (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘))))))) | ||
Theorem | mamuval 20927* | Multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃))) ⇒ ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))))) | ||
Theorem | mamufv 20928* | A cell in the multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃))) & ⊢ (𝜑 → 𝐼 ∈ 𝑀) & ⊢ (𝜑 → 𝐾 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐼(𝑋𝐹𝑌)𝐾) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾))))) | ||
Theorem | mamudm 20929 | The domain of the matrix multiplication function. (Contributed by AV, 10-Feb-2019.) |
⊢ 𝐸 = (𝑅 freeLMod (𝑀 × 𝑁)) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑃)) & ⊢ 𝐶 = (Base‘𝐹) & ⊢ × = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin)) → dom × = (𝐵 × 𝐶)) | ||
Theorem | mamufacex 20930 | Every solution of the equation 𝐴∗𝑋 = 𝐵 for matrices 𝐴 and 𝐵 is a matrix. (Contributed by AV, 10-Feb-2019.) |
⊢ 𝐸 = (𝑅 freeLMod (𝑀 × 𝑁)) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑃)) & ⊢ 𝐶 = (Base‘𝐹) & ⊢ × = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) & ⊢ 𝐺 = (𝑅 freeLMod (𝑀 × 𝑃)) & ⊢ 𝐷 = (Base‘𝐺) ⇒ ⊢ (((𝑀 ≠ ∅ ∧ 𝑃 ≠ ∅) ∧ (𝑅 ∈ 𝑉 ∧ 𝑌 ∈ 𝐷) ∧ (𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin)) → ((𝑋 × 𝑍) = 𝑌 → 𝑍 ∈ 𝐶)) | ||
Theorem | mamures 20931 | Rows in a matrix product are functions only of the corresponding rows in the left argument. (Contributed by SO, 9-Jul-2018.) |
⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) & ⊢ 𝐺 = (𝑅 maMul 〈𝐼, 𝑁, 𝑃〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ Fin) & ⊢ (𝜑 → 𝐼 ⊆ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃))) ⇒ ⊢ (𝜑 → ((𝑋𝐹𝑌) ↾ (𝐼 × 𝑃)) = ((𝑋 ↾ (𝐼 × 𝑁))𝐺𝑌)) | ||
Theorem | mndvcl 20932 | Tuple-wise additive closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼)) → (𝑋 ∘f + 𝑌) ∈ (𝐵 ↑m 𝐼)) | ||
Theorem | mndvass 20933 | Tuple-wise associativity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mnd ∧ (𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼) ∧ 𝑍 ∈ (𝐵 ↑m 𝐼))) → ((𝑋 ∘f + 𝑌) ∘f + 𝑍) = (𝑋 ∘f + (𝑌 ∘f + 𝑍))) | ||
Theorem | mndvlid 20934 | Tuple-wise left identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 0 = (0g‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → ((𝐼 × { 0 }) ∘f + 𝑋) = 𝑋) | ||
Theorem | mndvrid 20935 | Tuple-wise right identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 0 = (0g‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → (𝑋 ∘f + (𝐼 × { 0 })) = 𝑋) | ||
Theorem | grpvlinv 20936 | Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → ((𝑁 ∘ 𝑋) ∘f + 𝑋) = (𝐼 × { 0 })) | ||
Theorem | grpvrinv 20937 | Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-Sep-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → (𝑋 ∘f + (𝑁 ∘ 𝑋)) = (𝐼 × { 0 })) | ||
Theorem | mhmvlin 20938 | Tuple extension of monoid homomorphisms. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ ⨣ = (+g‘𝑁) ⇒ ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼)) → (𝐹 ∘ (𝑋 ∘f + 𝑌)) = ((𝐹 ∘ 𝑋) ∘f ⨣ (𝐹 ∘ 𝑌))) | ||
Theorem | ringvcl 20939 | Tuple-wise multiplication closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼)) → (𝑋 ∘f · 𝑌) ∈ (𝐵 ↑m 𝐼)) | ||
Theorem | mamucl 20940 | Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃))) ⇒ ⊢ (𝜑 → (𝑋𝐹𝑌) ∈ (𝐵 ↑m (𝑀 × 𝑃))) | ||
Theorem | mamuass 20941 | Matrix multiplication is associative, see also statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑂 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑂))) & ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑂 × 𝑃))) & ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑂〉) & ⊢ 𝐺 = (𝑅 maMul 〈𝑀, 𝑂, 𝑃〉) & ⊢ 𝐻 = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) & ⊢ 𝐼 = (𝑅 maMul 〈𝑁, 𝑂, 𝑃〉) ⇒ ⊢ (𝜑 → ((𝑋𝐹𝑌)𝐺𝑍) = (𝑋𝐻(𝑌𝐼𝑍))) | ||
Theorem | mamudi 20942 | Matrix multiplication distributes over addition on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑂〉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑂 ∈ Fin) & ⊢ + = (+g‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂))) ⇒ ⊢ (𝜑 → ((𝑋 ∘f + 𝑌)𝐹𝑍) = ((𝑋𝐹𝑍) ∘f + (𝑌𝐹𝑍))) | ||
Theorem | mamudir 20943 | Matrix multiplication distributes over addition on the right. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑂〉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑂 ∈ Fin) & ⊢ + = (+g‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑂))) & ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂))) ⇒ ⊢ (𝜑 → (𝑋𝐹(𝑌 ∘f + 𝑍)) = ((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍))) | ||
Theorem | mamuvs1 20944 | Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑂〉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑂 ∈ Fin) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂))) ⇒ ⊢ (𝜑 → ((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍) = (((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍))) | ||
Theorem | mamuvs2 20945 | Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.) |
⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑂〉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑂 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂))) ⇒ ⊢ (𝜑 → (𝑋𝐹(((𝑁 × 𝑂) × {𝑌}) ∘f · 𝑍)) = (((𝑀 × 𝑂) × {𝑌}) ∘f · (𝑋𝐹𝑍))) | ||
In the following, the square matrix algebra is defined as extensible structure Mat. In this subsection, however, only square matrices and their basic properties are regarded. This includes showing that (𝑁 Mat 𝑅) is a left module, see matlmod 20968. That (𝑁 Mat 𝑅) is a ring and an associative algebra is shown in the next subsection, after theorems about the identity matrix are available. Nevertheless, (𝑁 Mat 𝑅) is called "matrix ring" or "matrix algebra" already in this subsection. | ||
Syntax | cmat 20946 | Syntax for the square matrix algebra. |
class Mat | ||
Definition | df-mat 20947* | Define the algebra of n x n matrices over a ring r. (Contributed by Stefan O'Rear, 31-Aug-2015.) |
⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet 〈(.r‘ndx), (𝑟 maMul 〈𝑛, 𝑛, 𝑛〉)〉)) | ||
Theorem | matbas0pc 20948 | There is no matrix with a proper class either as dimension or as underlying ring. (Contributed by AV, 28-Dec-2018.) |
⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅) | ||
Theorem | matbas0 20949 | There is no matrix for a not finite dimension or a proper class as the underlying ring. (Contributed by AV, 28-Dec-2018.) |
⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅) | ||
Theorem | matval 20950 | Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) & ⊢ · = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet 〈(.r‘ndx), · 〉)) | ||
Theorem | matrcl 20951 | Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) | ||
Theorem | matbas 20952 | The matrix ring has the same base set as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Base‘𝐺) = (Base‘𝐴)) | ||
Theorem | matplusg 20953 | The matrix ring has the same addition as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (+g‘𝐺) = (+g‘𝐴)) | ||
Theorem | matsca 20954 | The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Scalar‘𝐺) = (Scalar‘𝐴)) | ||
Theorem | matvsca 20955 | The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝐴)) | ||
Theorem | mat0 20956 | The matrix ring has the same zero as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (0g‘𝐺) = (0g‘𝐴)) | ||
Theorem | matinvg 20957 | The matrix ring has the same additive inverse as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (invg‘𝐺) = (invg‘𝐴)) | ||
Theorem | mat0op 20958* | Value of a zero matrix as operation. (Contributed by AV, 2-Dec-2018.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 0 )) | ||
Theorem | matsca2 20959 | The scalars of the matrix ring are the underlying ring. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑅 = (Scalar‘𝐴)) | ||
Theorem | matbas2 20960 | The base set of the matrix ring as a set exponential. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 16-Dec-2018.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝐾 ↑m (𝑁 × 𝑁)) = (Base‘𝐴)) | ||
Theorem | matbas2i 20961 | A matrix is a function. (Contributed by Stefan O'Rear, 11-Sep-2015.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (𝐾 ↑m (𝑁 × 𝑁))) | ||
Theorem | matbas2d 20962* | The base set of the matrix ring as a mapping operation. (Contributed by Stefan O'Rear, 11-Jul-2018.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝐶 ∈ 𝐾) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ 𝐵) | ||
Theorem | eqmat 20963* | Two square matrices of the same dimension are equal if they have the same entries. (Contributed by AV, 25-Sep-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑋𝑗) = (𝑖𝑌𝑗))) | ||
Theorem | matecl 20964 | Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring. (Contributed by AV, 16-Dec-2018.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝑅) ⇒ ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐽) ∈ 𝐾) | ||
Theorem | matecld 20965 | Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring, deduction form. (Contributed by AV, 27-Nov-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ (𝜑 → 𝐼 ∈ 𝑁) & ⊢ (𝜑 → 𝐽 ∈ 𝑁) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼𝑀𝐽) ∈ 𝐾) | ||
Theorem | matplusg2 20966 | Addition in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ ✚ = (+g‘𝐴) & ⊢ + = (+g‘𝑅) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ✚ 𝑌) = (𝑋 ∘f + 𝑌)) | ||
Theorem | matvsca2 20967 | Scalar multiplication in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝐴) & ⊢ × = (.r‘𝑅) & ⊢ 𝐶 = (𝑁 × 𝑁) ⇒ ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) = ((𝐶 × {𝑋}) ∘f × 𝑌)) | ||
Theorem | matlmod 20968 | The matrix ring is a linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod) | ||
Theorem | matgrp 20969 | The matrix ring is a group. (Contributed by AV, 21-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Grp) | ||
Theorem | matvscl 20970 | Closure of the scalar multiplication in the matrix ring. (lmodvscl 19582 analog.) (Contributed by AV, 27-Nov-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐴) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵)) → (𝐶 · 𝑋) ∈ 𝐵) | ||
Theorem | matsubg 20971 | The matrix ring has the same addition as its underlying group. (Contributed by AV, 2-Aug-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (-g‘𝐺) = (-g‘𝐴)) | ||
Theorem | matplusgcell 20972 | Addition in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ ✚ = (+g‘𝐴) & ⊢ + = (+g‘𝑅) ⇒ ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 ✚ 𝑌)𝐽) = ((𝐼𝑋𝐽) + (𝐼𝑌𝐽))) | ||
Theorem | matsubgcell 20973 | Subtraction in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑆 = (-g‘𝐴) & ⊢ − = (-g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋𝑆𝑌)𝐽) = ((𝐼𝑋𝐽) − (𝐼𝑌𝐽))) | ||
Theorem | matinvgcell 20974 | Additive inversion in the matrix ring is cell-wise. (Contributed by AV, 17-Nov-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = (invg‘𝑅) & ⊢ 𝑊 = (invg‘𝐴) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑊‘𝑋)𝐽) = (𝑉‘(𝐼𝑋𝐽))) | ||
Theorem | matvscacell 20975 | Scalar multiplication in the matrix ring is cell-wise. (Contributed by AV, 7-Aug-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝐴) & ⊢ × = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 · 𝑌)𝐽) = (𝑋 × (𝐼𝑌𝐽))) | ||
Theorem | matgsum 20976* | Finite commutative sums in a matrix algebra are taken componentwise. (Contributed by AV, 26-Sep-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝐽 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈) ∈ 𝐵) & ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐴 Σg (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)))) | ||
The main result of this subsection are the theorems showing that (𝑁 Mat 𝑅) is a ring (see matring 20982) and an associative algebra (see matassa 20983). Additionally, theorems for the identity matrix and transposed matrices are provided. | ||
Theorem | matmulr 20977 | Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ · = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → · = (.r‘𝐴)) | ||
Theorem | mamumat1cl 20978* | The identity matrix (as operation in maps-to notation) is a matrix. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 )) & ⊢ (𝜑 → 𝑀 ∈ Fin) ⇒ ⊢ (𝜑 → 𝐼 ∈ (𝐵 ↑m (𝑀 × 𝑀))) | ||
Theorem | mat1comp 20979* | The components of the identity matrix (as operation in maps-to notation). (Contributed by AV, 22-Jul-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 )) & ⊢ (𝜑 → 𝑀 ∈ Fin) ⇒ ⊢ ((𝐴 ∈ 𝑀 ∧ 𝐽 ∈ 𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 )) | ||
Theorem | mamulid 20980* | The identity matrix (as operation in maps-to notation) is a left identity (for any matrix with the same number of rows). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 )) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑀, 𝑁〉) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) ⇒ ⊢ (𝜑 → (𝐼𝐹𝑋) = 𝑋) | ||
Theorem | mamurid 20981* | The identity matrix (as operation in maps-to notation) is a right identity (for any matrix with the same number of columns). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 )) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ 𝐹 = (𝑅 maMul 〈𝑁, 𝑀, 𝑀〉) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑁 × 𝑀))) ⇒ ⊢ (𝜑 → (𝑋𝐹𝐼) = 𝑋) | ||
Theorem | matring 20982 | Existence of the matrix ring, see also the statement in [Lang] p. 504: "For a given integer n > 0 the set of square n x n matrices form a ring." (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) | ||
Theorem | matassa 20983 | Existence of the matrix algebra, see also the statement in [Lang] p. 505: "Then Matn(R) is an algebra over R" . (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ AssAlg) | ||
Theorem | matmulcell 20984* | Multiplication in the matrix ring for a single cell of a matrix. (Contributed by AV, 17-Nov-2019.) (Revised by AV, 3-Jul-2022.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ × = (.r‘𝐴) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 × 𝑌)𝐽) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝐽))))) | ||
Theorem | mpomatmul 20985* | Multiplication of two N x N matrices given in maps-to notation. (Contributed by AV, 29-Oct-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ × = (.r‘𝐴) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ 𝑋 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝐶) & ⊢ 𝑌 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝐸) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝐸 ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑘 = 𝑖 ∧ 𝑚 = 𝑗)) → 𝐷 = 𝐶) & ⊢ ((𝜑 ∧ (𝑚 = 𝑖 ∧ 𝑙 = 𝑗)) → 𝐹 = 𝐸) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁 ∧ 𝑚 ∈ 𝑁) → 𝐷 ∈ 𝑈) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝐹 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑅 Σg (𝑚 ∈ 𝑁 ↦ (𝐷 · 𝐹))))) | ||
Theorem | mat1 20986* | Value of an identity matrix, see also the statement in [Lang] p. 504: "The unit element of the ring of n x n matrices is the matrix In ... whose components are equal to 0 except on the diagonal, in which case they are equal to 1.". (Contributed by Stefan O'Rear, 7-Sep-2015.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))) | ||
Theorem | mat1ov 20987 | Entries of an identity matrix, deduction form. (Contributed by Stefan O'Rear, 10-Jul-2018.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑁) & ⊢ (𝜑 → 𝐽 ∈ 𝑁) & ⊢ 𝑈 = (1r‘𝐴) ⇒ ⊢ (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, 1 , 0 )) | ||
Theorem | mat1bas 20988 | The identity matrix is a matrix. (Contributed by AV, 15-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 1 = (1r‘(𝑁 Mat 𝑅)) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 1 ∈ 𝐵) | ||
Theorem | matsc 20989* | The identity matrix multiplied with a scalar. (Contributed by Stefan O'Rear, 16-Jul-2018.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝐴) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → (𝐿 · (1r‘𝐴)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 𝐿, 0 ))) | ||
Theorem | ofco2 20990 | Distribution law for the function operation and the composition of functions. (Contributed by Stefan O'Rear, 17-Jul-2018.) |
⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻))) | ||
Theorem | oftpos 20991 | The transposition of the value of a function operation for two functions is the value of the function operation for the two functions transposed. (Contributed by Stefan O'Rear, 17-Jul-2018.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → tpos (𝐹 ∘f 𝑅𝐺) = (tpos 𝐹 ∘f 𝑅tpos 𝐺)) | ||
Theorem | mattposcl 20992 | The transpose of a square matrix is a square matrix of the same size. (Contributed by SO, 9-Jul-2018.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ 𝐵) | ||
Theorem | mattpostpos 20993 | The transpose of the transpose of a square matrix is the square matrix itself. (Contributed by SO, 17-Jul-2018.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ (𝑀 ∈ 𝐵 → tpos tpos 𝑀 = 𝑀) | ||
Theorem | mattposvs 20994 | The transposition of a matrix multiplied with a scalar equals the transposed matrix multiplied with the scalar, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 17-Jul-2018.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝐴) ⇒ ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → tpos (𝑋 · 𝑌) = (𝑋 · tpos 𝑌)) | ||
Theorem | mattpos1 20995 | The transposition of the identity matrix is the identity matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → tpos 1 = 1 ) | ||
Theorem | tposmap 20996 | The transposition of an I X J -matrix is a J X I -matrix, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.) |
⊢ (𝐴 ∈ (𝐵 ↑m (𝐼 × 𝐽)) → tpos 𝐴 ∈ (𝐵 ↑m (𝐽 × 𝐼))) | ||
Theorem | mamutpos 20997 | Behavior of transposes in matrix products, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.) |
⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) & ⊢ 𝐺 = (𝑅 maMul 〈𝑃, 𝑁, 𝑀〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃))) ⇒ ⊢ (𝜑 → tpos (𝑋𝐹𝑌) = (tpos 𝑌𝐺tpos 𝑋)) | ||
Theorem | mattposm 20998 | Multiplying two transposed matrices results in the transposition of the product of the two matrices. (Contributed by Stefan O'Rear, 17-Jul-2018.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = (.r‘𝐴) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → tpos (𝑋 · 𝑌) = (tpos 𝑌 · tpos 𝑋)) | ||
Theorem | matgsumcl 20999* | Closure of a group sum over the diagonal coefficients of a square matrix over a commutative ring. (Contributed by AV, 29-Dec-2018.) (Proof shortened by AV, 23-Jul-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑈 = (mulGrp‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑈 Σg (𝑟 ∈ 𝑁 ↦ (𝑟𝑀𝑟))) ∈ (Base‘𝑅)) | ||
Theorem | madetsumid 21000* | The identity summand in the Leibniz' formula of a determinant for a square matrix over a commutative ring. (Contributed by AV, 29-Dec-2018.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑈 = (mulGrp‘𝑅) & ⊢ 𝑌 = (ℤRHom‘𝑅) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑃 = ( I ↾ 𝑁)) → (((𝑌 ∘ 𝑆)‘𝑃) · (𝑈 Σg (𝑟 ∈ 𝑁 ↦ ((𝑃‘𝑟)𝑀𝑟)))) = (𝑈 Σg (𝑟 ∈ 𝑁 ↦ (𝑟𝑀𝑟)))) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |