Question

Give an example of a matrixAsuch thatim(A)is spanned by the vector$\left[\begin{array}{c}1\\ 5\end{array}\right]$.

Short answer

$\left[\begin{array}{c}1\\ 5\end{array}\right]$ is a matrix A such that im(A) is spanned by the vector$\left[\begin{array}{c}1\\ 5\end{array}\right]$ .

Step-by-step solution

Consider the parameters.

The image of a function consists of all the values the function takes in its target space. If f is a function from X to Y, then

$$\begin{gathered} image\left(f\right)=\left\{f\left(x\right):xinX\right\} \\ =\left\{binY:b=f\left(x\right),forsomexinX\right\} \end{gathered}$$

The span of A is the collection of all products of the form Ab for some compatible column vector $b$(i.e. b has as many elements as A has columns).

As $\left[\begin{array}{c}1\\ 5\end{array}\right]$has only one column, thus, $\left[\begin{array}{c}1\\ 5\end{array}\right]$is the matrix that is spanned by the vector $\left[\begin{array}{c}1\\ 5\end{array}\right]$.

Final answer.

$\left[\begin{array}{c}1\\ 5\end{array}\right]$ is a matrix A such that im(A) is spanned by the vector $\left[\begin{array}{c}1\\ 5\end{array}\right]$.