A linear transformation $T\colon\mathbb{R}^3\to\mathbb{R}^2$ whose matrix is
$$\left(\begin{array}{ccc}
1 & 3 & 3\\
2 & 6 & -3.5+k
\end{array}\right)$$
is onto if and only if $k\neq$__________
I'm a little confused by the notation here, so is the matrix given here supposed to be the matrix $A$ such that $XA \Rightarrow Y$? And what does the $\mathbb{R}^3$ and $\mathbb{R}^2$ notation mean? Does that mean a $3\times 3$ matrix and a $2\times 2$ matrix respectively?
Best Answer
$T$ takes in a column vector $(a_1, a_2, a_3)^T$, i.e. an element of $\mathbf R^3$, and sends it to \[ \begin{pmatrix} 1 & 3 & 3 \\ 2 & 6 & -3.5 + k \end{pmatrix} \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}. \] Convince yourself that this results in a $2 \times 1$ matrix, i.e. an element of $\mathbf R^2$.
For the surjectivity: the image of the transformation is the span of the columns of the matrix (why?). You also know that $\mathbf R^2$ is spanned by any two non-zero vectors that are not parallel. What is the span of the first two columns of the given matrix?