Exercise 6

Determine whether the following statements are true or false and provide a justification for your response.

1. A matrix transformation $T:\real^4\to\real^5$ is defined by $T(\xvec) = A\xvec$ where $A$ is a $4\times5$ matrix.

2. If $T:\real^3\to\real^2$ is a matrix transformation, then there are infinitely many vectors $\xvec$ such that $T(\xvec) = \zerovec$.

3. If $T:\real^2\to\real^3$ is a matrix transformation, then it is possible that every equation $T(\xvec) = \bvec$ has a solution for every vector $\bvec$.

4. If $T:\real^n\to\real^m$ is a matrix transformation, then the equation $T(\xvec) = \zerovec$ always has a solution.