# A linear transformation T &#x003A;<!-- : --> <mi mathvariant="double-struck">R 3

A linear transformation $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{2}$ whose matrix is
$\left(\begin{array}{ccc}1& 3& 3\\ 2& 6& -3.5+k\end{array}\right)$
is onto if and only if $k\ne$
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grcalia1
$T$ takes in a column vector $\left({a}_{1},{a}_{2},{a}_{3}{\right)}^{T}$, i.e. an element of ${\mathbf{R}}^{3}$, and sends it to
$\left(\begin{array}{ccc}1& 3& 3\\ 2& 6& -3.5+k\end{array}\right)\left(\begin{array}{c}{a}_{1}\\ {a}_{2}\\ {a}_{3}\end{array}\right).$
Convince yourself that this results in a $2×1$ matrix, i.e. an element of ${\mathbf{R}}^{2}$