Assume h: V rightarrow W is a linear map between the vector spaces V and W. Assu

Question

Assume h: V rightarrow W is a linear map between the vector spaces V and W. Assume also that the vectors {v_1,v_2,...v_n} are linearly dependent. We will to prove that the set of vectors (h(v_1),h(v_2),...h(v_n)} are also linearly dependent. The following steps will help you do that. What do you know about the set of vectors {v_1, V2,... v_n}? (write it in terms of a linear combination) Apply the function h to both sides of the linear combination you wrote in (a). Use the fact that h is a linear map (preserves structure) to re-write what you have above. State your conclusion. If we know that the set of vectors (h(x_1), h(x_2),... h(x_m)} are linearly independent, can the vectors {x_1, x_2,... x_m} be linearly dependent? Why?

Explanation / Answer

a)

We know that if :

a1 v1+....+an vn=0

Then not all ai s are 0. So for some j, aj is non zero

So we can write

ajvj=- sum_{k not j} ak vk

or

vj=-sum_{k not j} ak/aj vk

b)

Applying h gives

h(vj)=h(-sum_{k not j} ak/aj vk)

c)

Using linearity of h gives

h(vj)=-sum_{k not j} ak/aj h(vk)

d)

Hence,

h(v1).....,h(vn) are linearly dependent

d)

No. We need h to be injective for that.

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