# Linear transformation defined by a basis

I want to be sure that the follwing statement is true.

Let $$V, W$$ be $$\mathbb K$$-vector spaces. Let $$\{v_1, \ldots, v_n\}$$ be a basis of $$V$$ and $$\{w_1,\ldots,w_m\}$$ ($$m$$ can be different from $$n$$) be an arbitrary set of vectors of $$W$$. Then there exists a unique linear map $$F:V\longrightarrow W$$ such that $$F(v_i)=w_i$$ for every $$i=1,\ldots,n$$.

Is it true? I am not sure that $$m$$ can be different from $$n$$...

Thank you

If you let $$n=m$$, the statement is true, and this is a standard argument. If $$n, then you're essentially ignoring the last few vectors $$w_{n+1},\dots, w_m$$, so again it reduces to the case $$n=m$$. The final case $$n>m$$ doesn't make sense because if $$n>m$$, then the vector $$w_n$$ isn't even defined.
• Ok, thank you. So, in the finale case $n>m$ do not exist linear maps? Commented Nov 10, 2020 at 13:35
• @Redeldio the case $n>m$ is illogical, and the reason has nothing to do with linear algebra. Just think about it: you're trying to say $T(v_1)=w_1, \dots, T(v_m)=w_m$ and $T(v_{m+1})=w_{m+1}, \dots, T(v_n)=w_n$. But in your statement you only defined $w_1,\dots, w_m$, so what are $w_{m+1},\dots, w_n$? They have no meaning at all, which is why the statement itself is not well-formulated for $n>m$. This has as much meaning as saying $1=\text{spongebob}-\text{patrick}$. Commented Nov 10, 2020 at 13:37
• Ok, so, if I have a set of four vectors $\{v_1, \ldots, v_4\}$ in $V$ which are linearly independent and a set of three vectors $\{w_1,\ldots,w_3\}$ in $W$ which are linearly independent. How many linear maps $F$ from $V$ to $W$ there exist such that $F(v_i)=w_i$ for every $i$? Commented Nov 10, 2020 at 13:42
• Again that makes no sense. When you say "$F(v_i)=w_i$ for every $i$", what you mean is $F(v_1)=w_1, F(v_2)=w_2, F(v_3)=w_3$ AND $F(v_4)=w_4$. BUT, what is $w_4$ even? There is no $w_4$ at all in your problem statement. So, asking the question "how many linear maps...?" is completely meaningless. On the other hand if your question is "How many linear maps $F:V\to W$ are there such that $F(v_i)=w_i$ for every $1\leq i \leq 3$?", then this is a completely different question and this is meaningful. If this is the question, and the space $W$ has infinitely many elements then the answer is $\infty$. Commented Nov 10, 2020 at 13:45
• Man, this is a linear algebra exercise. I ask here because, as you, I don't understand it. My idea was to define $F(v_4)$ freely, for example $F(v_4)=0$. Then, there exist infinite linear maps. Can be this argument right? Commented Nov 10, 2020 at 13:47