Exercise2

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 \(T(\xvec) = \zerovec\text{.}\)

  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 \(\bvec\text{.}\)

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

  5. If \(T:\real^n\to\real^m\) is a matrix transformation and \(\vvec\) and \(\wvec\) two vectors in \(\real^n\text{,}\) then the vectors \(T(\vvec + t\wvec)\) form a line in \(\real^m\text{.}\)

in-context