Find matrix of change of basis from $(e_1, ... e_n)$ to $(e_n, e_{n-1} ... e_1)$, and $(e_1, e_2)$ to $(e_1 + e_2, e_1 - e_2)$

Question

a) Find the matrix of the change of basis from $K^n$, if the old basis is the standard basis $(e_1, ... e_n)$ and the new basis is $(e_n, e_{n-1} ... e_1)$

b) Find the matrix that describes the change from the old basis $(e_1, e_2)$ of $K^2$ to the new basis $(e_1 + e_2, e_1 - e_2)$

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a) $(e_1, ... e_n)\cdot \begin{bmatrix} 0 & 0 & \cdots & 0 & 1 \\ \vdots & \vdots & \ddots & 1 & 0 \\ \vdots & \vdots & 1 & \ddots & \vdots \\ 0 & 1 & 0 & \vdots & \vdots \\ 1 & 0 & 0 & 0 & 0 \end{bmatrix} = (e_n, e_{n-1} ... e_1)$

b)$(e_1, e_2)\cdot \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} = (e_1 + e_2, e_1 - e_2)$

Is this correct ? I'm quite new to the subject so your feedback would be really helpful

Thanks for your suppport !

Answer 1

If you have Basis $B = (b_1,b_2,..,b_n),$ and Basis $C = (c_1,c_2,...,c_n)$ of some vector space and want to find the basis change matrix from $B$ to $C$ what u do is: find the "coordinate vectors" of $(b_1,...,b_n)$ with regards to Basis $C$. that means: write $b_1$ as a linear combination of $c_1,...,c_n$ say $b_1 = a_1c_1 + ... + a_nc_n$ then $(a_1,...,a_n)$ ist your coordinate vector of $b_1$ with regards to basis $C$.(This is unique which follows directly from def. of basis) You do that for every basis vector of $B$ so u end with n row vectors. U transpose all of these vectors and write them in a matrix such that e.g. $(a_1,...,a_n)$ is the first column of ur matrix. (This might not be the best explanation so heres an example.

In your question (b) we have got the following:

$e_1 = 1/2(e_1+e_2) + 1/2(e_1-e_2)$

$e_2 = 1/2(e_1+e_2) - 1/2(e_1-e_2)$

resulting in the matrix

$\begin{pmatrix} 1/2 & 1/2 \\ 1/2 & -1/2 \end{pmatrix}$

Of course I apologize for saying ur b) is correct bcs it is not but ur a) is correct.