[Math] Condition for mapping linearly independent vectors to linearly independent vectors

linear algebra

Let $f \colon U \to V$ be a linear map and $\{u_1, u_2, \ldots, u_n\}$ be the set of linearly independent vectors in $U$.

Then the set $\{f(u_1), f(u_2),\ldots,f(u_n)\}$ is linearly independent iff

a. $f$ is one-one & onto

b. $f$ is one-one

c. $f$ is onto

d. $U = V$

I came to conclusion that $f$ has to be one-one.
This is what I did.

We have $$a_1 u_1 + a_2 u_2+ \ldots + a_n u_n = 0.$$

For $\{f(u_1), f(u_2),\ldots,f(u_n)\}$ to be independent, we have to have $$a_1 f(u_1) + a_2 f(u_2) + \cdots + a_n f(u_n) = 0.$$

Simplifying we get $$f(a_1 u_1 + a_2 u_2 + \cdots + a_n u_n) = 0$$
i.e. $f(0)=0$.

This implies that $\ker f = \{0\}$ i.e. $f$ has to be one-one.

Is my logic correct? Also what's the correct answer and why?

Best Answer

In the light of @Gerry's comment, take the following linear transformation: $$T:\mathbb R^3\to\mathbb R^2$$ $$T(a_1,a_2,a_3)=(a_1-a_2,2a_3)$$ $N(T)=\{(a,a,0)\mid a\in\mathbb R\}$ and $R(T)=\mathbb R^2$. So a,b,d is wrong. Search for another for c.

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