Question

Suppose TA : R^2 --> R^2 is defined so that TA(x) = 0 for all x E R3. Show that A is the 2 x 2 zero matrix.

Answer 1

Observe that the set
`B=\{(1,0), (0,1)\}` is a basis for
`\mathbb{R}^2`, then the matrix is defined by the image of the elements of B.

Since
`T(1,0)=(0,0)` and
`T(0,1)=(0,0)` then the associated matrix to T is
`A=\left[\begin{array}{cc}T(1,0)^T & T(0,1)^T\end{array}\right]=\left[\begin{array}{cc}0&0\\0&0\end{array}\right]` that is the zero matrix.