Question
Show that no linear map $T: \mathbb{F}^{5} \rightarrow \mathbb{F}^{2}$ canhave as its null space the set$$\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right) \in \mathbb{F}^{5} \mid x_{1}=3 x_{2}, x_{3}=x_{4}=x_{5}\right\} .$$
Step 1
First, let's recall the definition of a linear map. A map $T: \mathbb{F}^5 \rightarrow \mathbb{F}^2$ is linear if for all $x, y \in \mathbb{F}^5$ and $c \in \mathbb{F}$, we have $T(x + y) = T(x) + T(y)$ and $T(cx) = cT(x)$. Show more…
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