Linear Transformations and Basis

[spratleyj]

Homework Statement

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

Homework Equations

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.

 

[micromass]

Homework Statement

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

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

we only need to show that $${T(v_1), ... , T(v_k)}$$ spans $$T(c_1v_1 + ... + c_kv_k)$$

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

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