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## Homework Statement

Find a basis for the following vector space:

## V = \{ p \in \mathbb C_{\leq4} ^{[z]} | \ p(1)=p(i) ## and ## p(2)=0 \} ##

(Where ## \mathbb C_{\leq4} ^{[z]} ## denotes the polynomials of degree at most 4)

## Homework Equations

N/A

## The Attempt at a Solution

I tried to find bi-terms with all possible degree combinations. such as:

## 8z-z^4 ## and ##2z^3-z^4 ##

The ## p(2)=0 ## part is easy, but I can't seem to find any bi-terms that pass ## p(1)=p(i) ##

I'm afraid that randomly trying out tri-term and quad-term combinations can get messy.

And a side question: Is it true that, suppose there are no polynomials for which ## p(1)=p(i) ##, or more generally, a vector space that is the trivial one which contains only the zero vector. Then the basis of that vector space is the empty set?

Could you please help me? Thanks :)

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