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Matrix

Last updated Nov 1, 2022

# Definition

Let SS be a Set and let m,nNm,n \in \mathbb{N}. We say (AijS)i[n],j[m](A_{ij} \in S)_{i \in [n], j \in [m]} is a Matrix

Matrix

Definition Let SS be a and let m,nNm,n \in \mathbb{N}. We say (AijS)i[n],j[m](A{ij} \in S){i \in [n], j \in [m]}...

11/7/2022

.

# Remarks

  1. We often represent a Matrix

    Matrix

    Definition Let SS be a and let m,nNm,n \in \mathbb{N}. We say (AijS)i[n],j[m](A{ij} \in S){i \in [n], j \in [m]}...

    11/7/2022

    as a “nn by mm” table. For example (123456)\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6\end{pmatrix} is a 2×32 \times 3 Matrix

    Matrix

    Definition Let SS be a and let m,nNm,n \in \mathbb{N}. We say (AijS)i[n],j[m](A{ij} \in S){i \in [n], j \in [m]}...

    11/7/2022

    over Z\mathbb{Z}.
  2. We refer to the Set of all n×mn \times m matrices

    Matrix

    Definition Let SS be a and let m,nNm,n \in \mathbb{N}. We say (AijS)i[n],j[m](A{ij} \in S){i \in [n], j \in [m]}...

    11/7/2022

    on SS as Sn×mS^{n \times m}.
  3. Oftentimes, SS is taken to be a Ring or Field

    Field

    Definition Suppose XX is a , and +:X×XX+: X \times X \to X, :X×XX: X \times X \to X are...

    11/7/2022

    . This allows us to define Matrix Multiplication.