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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?

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    $\mathbb{R}^n$ denotes [$n$-dimensional Euclidean space](http://en.wikipedia.org/wiki/Euclidean_space). The notation $T : \mathbb{R}^3 \to \mathbb{R}^2$ means that $T$ send a $3$-dimensional vector to a $2$-dimensional vector. A map $T : X \to Y$ is [onto](http://en.wikipedia.org/wiki/Surjective_function) if every element $y \in Y$ can be realized by a point $x \in X$ (I.e., for every element $y$ in $Y$, there is an element $x$ such that $T(x) = y$). The question wants you to find the value(s) of $k$ such that the transformation $T : \mathbb{R}^3 \to \mathbb{R}^2$ is onto.2012-03-05

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$T$ 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?

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    @StickFigs I believe that's the answer, yes.2012-03-05