This is an exercise I had on the Linear Algebra final exam I failed. I could only do a part and I'm still stuck for most of it. So I'll write it entirely and I'll ask help only for part 1) until we finish it, then ask for further help.

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Let

be the ordered basis of

where

,

,

.

Let

be defined by

,

,

.

1)Give an explicit description of

, calculate its dimension and give a basis.

2)Give an explicit description of

, calculate its dimension and give a basis.

3)Calculate

, where

.

4)If

, find the coordinates of

with respect to

.

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1) My attempt : only thoughts. I must find all the vectors

such that

. I'm at a loss. I don't even know if the transformation is linear. I've absolutely no idea about how to find the kernel of it.