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I am reading Gerard Walschap's book: "Multivariable Calculus and Differential Geometry" and am focused on Chapter 1: Euclidean Space ... ...

I have a further question concerning an aspect of the proof of Theorem 1.3.1 ...

Theorem 1.3.1 and its proof read as follows:

Equation 1.3.1 in the proof implies that

\(\displaystyle \text{det} ( v_1, \cdot \cdot \cdot , v_n ) = \text{det} \left( \ \begin{bmatrix} a_{ 11} \\ \ \\ . \\ . \\ . \\ \ \\ a_{n1} \end{bmatrix} , \cdot \cdot \cdot , \begin{bmatrix} a_{ 1n} \\ \ \\ . \\ . \\ . \\ \ \\ a_{nn} \end{bmatrix} \ \right) \)

so ... in other words ...

\(\displaystyle v_j = \begin{bmatrix} a_{ 1j} \\ \ \\ . \\ . \\ . \\ \ \\ a_{nj} \end{bmatrix}\)

Now I can see that \(\displaystyle v_j = a_{ 1j} e_1 + a_{ 2j} e_2 + \cdot \cdot \cdot + a_{nj} e_n \)

... and believe that to represent \(\displaystyle v_j\) as \(\displaystyle \begin{bmatrix} a_{ 1j} \\ \ \\ . \\ . \\ . \\ \ \\ a_{nj} \end{bmatrix}\)

... is just a convention to represent \(\displaystyle v_j\) by its coordinates with respect to the current basis ... ...

Can someone please confirm that this is the correct interpretation of the left hand side of equation 1.3.1 in the proof ...

Hope someone can help ...

Peter

I have a further question concerning an aspect of the proof of Theorem 1.3.1 ...

Theorem 1.3.1 and its proof read as follows:

Equation 1.3.1 in the proof implies that

\(\displaystyle \text{det} ( v_1, \cdot \cdot \cdot , v_n ) = \text{det} \left( \ \begin{bmatrix} a_{ 11} \\ \ \\ . \\ . \\ . \\ \ \\ a_{n1} \end{bmatrix} , \cdot \cdot \cdot , \begin{bmatrix} a_{ 1n} \\ \ \\ . \\ . \\ . \\ \ \\ a_{nn} \end{bmatrix} \ \right) \)

so ... in other words ...

\(\displaystyle v_j = \begin{bmatrix} a_{ 1j} \\ \ \\ . \\ . \\ . \\ \ \\ a_{nj} \end{bmatrix}\)

Now I can see that \(\displaystyle v_j = a_{ 1j} e_1 + a_{ 2j} e_2 + \cdot \cdot \cdot + a_{nj} e_n \)

... and believe that to represent \(\displaystyle v_j\) as \(\displaystyle \begin{bmatrix} a_{ 1j} \\ \ \\ . \\ . \\ . \\ \ \\ a_{nj} \end{bmatrix}\)

... is just a convention to represent \(\displaystyle v_j\) by its coordinates with respect to the current basis ... ...

Can someone please confirm that this is the correct interpretation of the left hand side of equation 1.3.1 in the proof ...

Hope someone can help ...

Peter

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