$For what values k is the system consistent?$

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$For what values k is the system consistent?$

Let $T : \mathbb{R} ^{2} \rightarrow \mathbb{R} ^{3}$ be a linear representation with a standard matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ -2 & -4 & -k \end{bmatrix} $
where k is a number that we can vary. (Note that it is not an extended matrix).

a) For what values k is the system $Ax = \begin{bmatrix}-17\\34\end{bmatrix} $ consistent?

b) For what values k is the system $Ax = \begin{bmatrix}1\\34\end{bmatrix} $ consistent?

c) For what values k is the linear representation T onto?

d) For what values k is the linear representation T one-to-one?

Linear Algebra

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I assume you mean $T \colon \mathbb{R}^3 \rightarrow \mathbb{R}^2$.

We have $Im(T) = \mathsf{span}\left( \begin{bmatrix} 1 \\ -2 \end{bmatrix}, \begin{bmatrix} 2 \\ -4 \end{bmatrix}, \begin{bmatrix} 3 \\ -k \end{bmatrix} \right) = \mathsf{span}\left( \begin{bmatrix} 1 \\ -2 \end{bmatrix}, \begin{bmatrix} 3 \\ -k \end{bmatrix} \right)$ as the second vector is a linear multiple of the first.

1. The vector $\begin{bmatrix} -17 \\ 34 \end{bmatrix}$ belongs to the image for every value of $k$, as it is $-17 \cdot \begin{bmatrix} 1 \\ -2 \end{bmatrix}$.

2. Since the vector $\begin{bmatrix} 1 \\ 34 \end{bmatrix}$ is not a linear multiple of $\begin{bmatrix} 1 \\ -2 \end{bmatrix}$, if belongs to the image if and only if the map is surjective, that is, whenever $\begin{bmatrix} 3 \\ -k \end{bmatrix}$ is not a linear multiple of $\begin{bmatrix} 1 \\ -2 \end{bmatrix}$, so whenever $k \neq 6$. In this case, in fact, the matrix has rank $2$ and so $T$ is surjective.

3. As explained in the second point, $T$ is surjective $\iff k \neq 6$.

4. $T$ is a map from a space of higher dimension to a space of lower dimension, so it is never injective, regardless of the value of $k$. Indeed, \[ T \begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix} = 0 \] for every value of $k$.

Alessandro Iraci

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$For what values k is the system consistent?$

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