What do the columns of the standard matrix of a linear map represent?

What size matrix represents a map \(T:\mathbb R^3\to\mathbb R^2\)?

Find the standard matrix for \(T(x,y)=(3x,5y)\).

For \(A=\begin{pmatrix}2&-1\\4&0\end{pmatrix}\), what are \(T(\mathbf e_1)\) and \(T(\mathbf e_2)\)?

Find the matrix of \(T(x,y)=(x+y,2x-y)\).

The matrix \(A=\begin{pmatrix}1&0&2\\0&3&-1\end{pmatrix}\) represents \(T:\mathbb R^3\to\mathbb R^2\). Find \(T(x,y,z)\).

A linear map satisfies \(T(1,0)=(2,-1,0)\) and \(T(0,1)=(4,3,5)\). Write its standard matrix.

For \(A=\begin{pmatrix}2&1\\-1&3\end{pmatrix}\), compute the output for input \((4,-2)\).

Find the standard matrix of \(T(x,y,z)=(x-z,2x+y,3z)\).

Given \(A=\begin{pmatrix}1&2\\3&4\\5&6\end{pmatrix}\), express \(T(x,y)\) as a linear combination of the columns.

A map has matrix \(A=\begin{pmatrix}1&1\\0&2\end{pmatrix}\). Verify from columns that \(T(3,4)=3T(1,0)+4T(0,1)\).

Explain why a \(3\times2\) matrix cannot represent a standard-coordinate map from \(\mathbb R^3\) to \(\mathbb R^2\).

A linear map sends \((1,1)\) to \((3,5)\) and \((1,-1)\) to \((1,-1)\). Find its standard matrix.

Find the matrix of the map that projects \((x,y,z)\) onto the \(xz\)-plane by removing the \(y\)-component.

For which \(a\) does \(A=\begin{pmatrix}1&a\\2&4\end{pmatrix}\) send \((1,-1)\) to \((0,-2)\)?

Find all \(b\) such that \(A=\begin{pmatrix}b&1\\1&b\end{pmatrix}\) sends both \((1,1)\) and \((1,-1)\) to scalar multiples of themselves.

A matrix has columns \(\mathbf c_1=(1,2)\), \(\mathbf c_2=(0,1)\), and \(\mathbf c_3=(3,-1)\). Find the input \((x,y,z)\) that maps to \(2\mathbf c_1-\mathbf c_2+4\mathbf c_3\).

A student writes the basis-vector images as rows instead of columns. Diagnose the error for \(T(1,0)=(2,3)\), \(T(0,1)=(5,7)\).

Prove that knowing a linear map on a basis determines its matrix and its value on every vector.

Can two different standard matrices represent the same linear map \(\mathbb R^2\to\mathbb R^2\)? Justify your answer.