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giraffe714

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- TL;DR Summary
- The formula for finding change of basis transformations is b'_j = c_1j b_1 + ... + c_nj v_n, but when the coefficients c_1j are put into a matrix and we multiply that matrix by b_1, it doesn't give back b'_1?

The formula my textbook provides for finding change of basis matrices is:

$$b'_j = a_{1j} b_1 + \cdots + a_{nj} b_n$$

I assume, since that's the convention and also because Wikipedia itself uses this formula like this, that the first index of the c's is the row, and the second is the columns. So, given some basis B and some basis B', we should be able to construct the matrix P which is a change of basis between the two. Let's also assume that B and B' each have 2 vectors for simplicity.

Hence, we can construct the matrix

$$ P = \begin{pmatrix} a_{11} && a_{12} \\ a_{21} && a_{22} \end{pmatrix} $$

All of this is standard. My confusion comes when I try to directly multiply this P by some b_1.

$$ \begin{pmatrix} a_{11} && a_{12} \\ a_{21} && a_{22} \end{pmatrix} \begin{pmatrix} b_{11} \\ b_{21} \end{pmatrix} = \begin{pmatrix} a_{11} b_{11} + a_{12} b_{21} \\ a_{21} b_{11} + a_{22} b_{21} \end{pmatrix} $$

Which isn't the formula ##b'_j = a_{1j} b_1 + \cdots + a_{nj} v_n##. Or at least, it's not the formulas ##b'_{j1} = a_{1j} b_{11} + \cdots + a_{nj} b_{n1}## and ##b'_{j2} = a_{1j} b_{12} + \cdots + a_{nj} b_{n2}##, which is surely just the above formula separated out by components, which should be allowed? Which step here is flawed? I can't find any explanation for this discrepancy online. By all means - the formula works when I use concrete bases. Is this level of abstraction simply not allowed in linear algebra? I understand that I'm making a mistake somewhere in this reasoning but I can't find where.

The weird thing is, this formula works. By all means, if I use two concrete bases, such as, for example:

B = (1, 0), (0, 1) and B' = (1, 0), (1, 1)

This formula works. But I don't know why.

$$b'_j = a_{1j} b_1 + \cdots + a_{nj} b_n$$

I assume, since that's the convention and also because Wikipedia itself uses this formula like this, that the first index of the c's is the row, and the second is the columns. So, given some basis B and some basis B', we should be able to construct the matrix P which is a change of basis between the two. Let's also assume that B and B' each have 2 vectors for simplicity.

Hence, we can construct the matrix

$$ P = \begin{pmatrix} a_{11} && a_{12} \\ a_{21} && a_{22} \end{pmatrix} $$

All of this is standard. My confusion comes when I try to directly multiply this P by some b_1.

$$ \begin{pmatrix} a_{11} && a_{12} \\ a_{21} && a_{22} \end{pmatrix} \begin{pmatrix} b_{11} \\ b_{21} \end{pmatrix} = \begin{pmatrix} a_{11} b_{11} + a_{12} b_{21} \\ a_{21} b_{11} + a_{22} b_{21} \end{pmatrix} $$

Which isn't the formula ##b'_j = a_{1j} b_1 + \cdots + a_{nj} v_n##. Or at least, it's not the formulas ##b'_{j1} = a_{1j} b_{11} + \cdots + a_{nj} b_{n1}## and ##b'_{j2} = a_{1j} b_{12} + \cdots + a_{nj} b_{n2}##, which is surely just the above formula separated out by components, which should be allowed? Which step here is flawed? I can't find any explanation for this discrepancy online. By all means - the formula works when I use concrete bases. Is this level of abstraction simply not allowed in linear algebra? I understand that I'm making a mistake somewhere in this reasoning but I can't find where.

The weird thing is, this formula works. By all means, if I use two concrete bases, such as, for example:

B = (1, 0), (0, 1) and B' = (1, 0), (1, 1)

This formula works. But I don't know why.

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