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)