Prove $$e_1=(1-i,3+2i), e_2=(-1+2i,3-i) \\ f_1=(1,9+3i),f_2=(-1+3i,9)$$ are bases of $\mathbb{C}^2$ and find $C_{e\rightarrow f}$. Find coordinates of vector in basis $e$ if there are coordinates $(x_1,x_2)$ in basis $f$. So I computed a rank of matrix: \begin{bmatrix}1-i & 3+2i\\-1+2i & 3-i \end{bmatrix} and the same for basis $f$. $\text{rank}(e)=2$ and $\text{rank}(f)=2$. But I don't know how to find matrix $C_{e\rightarrow f}$. Please help
Prove $e,f$ are bases of $\mathbb{C}^2$
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linear-algebra
vector-spaces
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0Figure out the coefficients so that you can write $e_1 = af_1 + bf_2$ and $e_2 = cf_1 + df_2$. Then your change of basis matrix will be $\pmatrix{a & c \\ b & d}$. – 2017-02-28
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0The problem is that I can't determine $a,b,c,d$. I study on y own and nobody can show me how to do it with complex numbers – 2017-02-28
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0Use your knowledge of complex arithmetic to solve $\begin{cases}1-i = a(1) +b(-1+3i) \\ 3+2i = a(9+3i)+b(9)\end{cases}$ to get $a$ and $b$. Find $c$ and $d$ similarly. Keep in mind though that $a,b,c,d$ are generally going to be complex numbers. – 2017-02-28