Matrices

Definition

An $m\times n$ matrix is a rectangular array with $m$ rows and $n$ columns.

$$A=(a_{j,k}{)}_{j=1,k=1}^{m,n}=\left(\begin{array}{cccc}{a}_{1,1}& a_{1,2}& \dots & a_{1,n}\\ a_{2,1}& a_{2,2}& \dots & a_{2,n}\\ ⋮& ⋮& & ⋮\\ a_{m,1}& a_{m,2}& \dots & a_{m,n}\end{array}\right)$$

Note

Matrices can be added and scaled, and the set of all $m\times n$ matrices satisfy all of the axioms of a vector space. Matrices can be treated as vectors.

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Systems of linear equations

Matrices can be used to represent systems of linear equations.
For example, these equations

$$\begin{array}{rl}& a_{11}{x}_1+a_{12}{x}_2+a_{13}{x}_3=b_1\\ & a_{21}{x}_1+a_{22}{x}_2+a_{23}{x}_3=b_2\\ & a_{31}{x}_1+a_{32}{x}_2+a_{33}{x}_3=b_3\end{array}$$

can be represented as

$$\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]$$ $$A\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]$$

The number of rows in $A$ represents the number of equations, and the number of columns in $A$ represents the number of variables. $A$ is called the coefficient matrix of the system.

Augmented matrix

The above representation can be further condensed into an augmented matrix:

$$\left[\begin{array}{cccc}{a}_{11}& a_{12}& a_{13}& b_1\\ a_{21}& a_{22}& a_{23}& b_2\\ a_{31}& a_{32}& a_{33}& b_3\end{array}\right]$$

Solving linear systems, Pivots

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Representations of linear transformations

Matrices can be used to represent linear transformations.

Definition 1(Definition).

A matrix is called invertible if the corresponding linear transformation is invertible.

This gives way to a definition of matrix multiplication.

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Transpose

Given a matrix $A$, its transpose is defined by transforming the rows of $A$ into columns.

$${\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right)}^T=\left(\begin{array}{cc}1& 4\\ 2& 5\\ 3& 6\end{array}\right)$$

When you take the transpose of a product, you change the order of terms.

$$(AB{)}^T=B^T{A}^T$$