Thread: Determing if F linear map

1. Determing if F linear map

Suppose that $F: \mathbb{R}^4 \to \mathbb{R}^2$ is a function satisfying

$F(\boldsymbol{a}) = \begin{pmatrix}3\\-1\end{pmatrix}, F(\boldsymbol{b}) = \begin{pmatrix}3\\0\end{pmatrix}, F(\boldsymbol{u}) = \begin{pmatrix}1\\-3\end{pmatrix}, F(\boldsymbol{v}) = \begin{pmatrix}4\\1\end{pmatrix}$ and $F(\boldsymbol{w}) = \begin{pmatrix}0\\4\end{pmatrix}$

Is F a linear map? Explain.

Im not sure how to do this.

2. What are a, b, u, v, and w? Have you got no information about these?

3. Originally Posted by HappyJoe
What are a, b, u, v, and w? Have you got no information about these?
No info was given.

4. Simply giving the results of F on 4 separate vectors is not enough to tell whether F is linear or not- it is not enough to determine what F is!

For example, it would be possible to define F to be exactly what is given on those four vectors and 0 for all other vectors. That would NOT be a linear map.

If we are given the result of applying F to four basis vectors, a, b, u, v, for $R^4$ and F is "defined by linearity" for all other vectors (that is, if a, b, u, v are basis vectors, then for any v in $R^4$, v= pa+ qb+ ru+ sv for some numbers p, q, r, and s, and then F(v)= pF(a)+ qF(b)+ rF(u)+ sF(v), then F is, of course, linear.