# Linear Transformations and Basis

1. Oct 18, 2011

### spratleyj

1. The problem statement, all variables and given/known data

Show that if $${ v_1, ... , v_k}$$ spans $$V$$ then $${T(v_1), ... , T(v_k)}$$ spans $$T(v)$$

2. Relevant equations

3. The attempt at a solution

So we know that every vector in V can be written as a linear combination of $$v_1,...v_k$$ thus we only need to show that $${T(v_1), ... , T(v_k)}$$ spans $$T(c_1v_1 + ... + c_kv_k)$$

However, I'm not sure how to do that.

2. Oct 18, 2011

### micromass

Staff Emeritus
You probably meant to write T(V) here...

What in earth does that even mean??????

You need to show that $\{T(v_1),...,T(v_k)\}$ spans T(V). So take a vector in T(V) and show that it can be written as

$$T(c_1v_1 + ... + c_kv_k)$$