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Nov 1, 2022
# DefinitionLet S S S be a Set and let m , n ∈ N m,n \in \mathbb{N} m , n ∈ N . We say ( A i j ∈ S ) i ∈ [ n ] , j ∈ [ m ] (A_{ij} \in S)_{i \in [n], j \in [m]} ( A ij ∈ S ) i ∈ [ n ] , j ∈ [ m ] is a Matrix
Matrix
Definition
Let S S S be a and let m , n ∈ N m,n \in \mathbb{N} m , n ∈ N . We say ( A i j ∈ S ) i ∈ [ n ] , j ∈ [ m ] (A{ij} \in S){i \in [n], j \in [m]} ( A ij ∈ S ) i ∈ [ n ] , j ∈ [ m ] ...
11/7/2022
.
We often represent a Matrix
Matrix
Definition
Let S S S be a and let m , n ∈ N m,n \in \mathbb{N} m , n ∈ N . We say ( A i j ∈ S ) i ∈ [ n ] , j ∈ [ m ] (A{ij} \in S){i \in [n], j \in [m]} ( A ij ∈ S ) i ∈ [ n ] , j ∈ [ m ] ...
11/7/2022
as a “n n n by m m m ” table. For example
( 1 2 3 4 5 6 ) \begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6\end{pmatrix} ( 1 4 2 5 3 6 )
is a 2 × 3 2 \times 3 2 × 3 Matrix
Matrix
Definition
Let S S S be a and let m , n ∈ N m,n \in \mathbb{N} m , n ∈ N . We say ( A i j ∈ S ) i ∈ [ n ] , j ∈ [ m ] (A{ij} \in S){i \in [n], j \in [m]} ( A ij ∈ S ) i ∈ [ n ] , j ∈ [ m ] ...
11/7/2022
over Z \mathbb{Z} Z . We refer to the Set of all n × m n \times m n × m matrices
Matrix
Definition
Let S S S be a and let m , n ∈ N m,n \in \mathbb{N} m , n ∈ N . We say ( A i j ∈ S ) i ∈ [ n ] , j ∈ [ m ] (A{ij} \in S){i \in [n], j \in [m]} ( A ij ∈ S ) i ∈ [ n ] , j ∈ [ m ] ...
11/7/2022
on S S S as S n × m S^{n \times m} S n × m . Oftentimes, S S S is taken to be a Ring or Field
Field
Definition
Suppose X X X is a , and + : X × X → X +: X \times X \to X + : X × X → X , : X × X → X : X \times X \to X : X × X → X are...
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. This allows us to define Matrix Multiplication .