# 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 $(e_1 + e_2, e_1 - e_2)$

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 $$(e_1 + e_2, e_1 - e_2)$$

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

• @GloriousErin Thank you for the edit, I just notice now that I wrote $(e_n, e_{n-1}... e_n)$ instead of $(e_n, e_{n-1} ... e_1)$ Commented Feb 27, 2023 at 9:18

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.

• How did you get $e_1 = 1/2(e_1+e_2) + 1/2(e_1-e_2)$ and $e_2 = 1/2(e_1+e_2) - 1/2(e_1-e_2)$ ? I don't quite see how you got the 1/2, is this linked to the change of basis ? Commented Feb 27, 2023 at 12:35
• The 1/2 just comes from basic calculation. As I already said, what u do is express the "old" basis vectors as a linear combination of the "new" basis and ur coefficients (written in the correct order and transposed etc..) build ur basis change matrix. Commented Feb 27, 2023 at 12:41