Let $[e_1,e_2,e_3]$ and $[ f_1, f_2, f_3]$ be ordered bases of a vector space $V$ over $\mathbb{R}$.

Question

Which of the following functions from $V$ to $V$ are linear maps? Using the definition of a linear map, justify your answers. (You may not assume that $V = \mathbb{R^3}$)

(a) $T(x_1e_1 + x_2e_2 + x_3e_3) = |x_1| f_1 + x_3 f_3$

(b) $S(x_1e_1 + x_2e_2 + x_3e_3) = (x_1 + x_2)f_1$

I've looked through my notes but I can't find anything that is in any way similar to this so really any guidance would be appreciated.

Answer 1

Hints:

We have $$Te_1 = T(-e_1) = f_1$$ Can $T$ be linear?

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Show that $$S(\alpha (x_1e_1 + x_2e_2 + x_3e_3) + \beta (y_1e_1 + y_2e_2 + y_3e_3) ) = \alpha S(x_1e_1 + x_2e_2 + x_3e_3) + \beta S(y_1e_1 + y_2e_2 + y_3e_3)$$

and conclude that $S$ is linear.