Chapter 1, Problem 1E

TRUE OR FALSE? ${}^{19}$

Determine whether the statements that follow are true or false, and justify your answer.

1. If A is an $n\times n$ matrix and $\stackrel{\to }{x}$ is a vector in ${ℝ}^n$ , then the product $A\stackrel{\to }{x}$ is a linear combination of the columns of matrix A.

To determine

To find: Whether the given statement is true of false.

Answer to Problem 1E

The given statement is true.

Explanation of Solution

Given information:

The statement is “If $A$ is an $n\times n$ matrix and $\overrightarrow{x}$ is a vector in ${ℝ}^n$ then product $A\overrightarrow{x}$ is a linear combination of the columns of the matrix $A$ .”

Consider the matrix $A$ .

  $\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& \cdots & a_{nn}\end{array}\right]$

Consider the matrix $\overrightarrow{x}$ .

  $\overrightarrow{x}=\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]$

Multiply above matrices.

  $\begin{array}{c}A\overrightarrow{x}=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& \cdots & a_{nn}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]\\ =\left[\begin{array}{cccc}{a}_{11}{x}_1& a_{12}{x}_2& \cdots & a_{1n}{x}_n\\ a_{21}{x}_1& a_{22}{x}_2& \cdots & a_{2n}{x}_n\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}{x}_1& a_{n2}{x}_2& \cdots & a_{nn}{x}_n\end{array}\right]\\ =x_1\left[\begin{array}{c}{a}_{11}\\ a_{21}\\ ⋮\\ a_{n1}\end{array}\right]+x_2\left[\begin{array}{c}{a}_{12}\\ a_{22}\\ ⋮\\ a_{n2}\end{array}\right]+\mathrm{...}+x_n\left[\begin{array}{c}{a}_{1n}\\ a_{2n}\\ ⋮\\ a_{nn}\end{array}\right]\end{array}$

Therefore, the given statement is true.