Let T be a linear transformation from $\mathbb{R}^4 \rightarrow \mathbb{R}^5$.

$$T \begin{pmatrix} x_1\\ x_2\\ x_3\\ x_4 \end{pmatrix} = \begin{pmatrix} x_1\\ x_2\\ x_3\\ x_4\\ 1 \end{pmatrix} $$

Is T a valid transformation, now that the 5th coordianate is a constant or does it have to be expressed by $x_1, x_2, x_3$ or $x_4$?

Thanks in advance.


The function $T$ is not a linear transformation, as given, since the zero vector is not in its range.


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