- #1

etotheipi

2019 Award

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## Main Question or Discussion Point

Consider the transformation of the components of a vector ##\vec{v}## from an orthonormal coordinate system with a basis ##\{\vec{e}_1, \vec{e}_2, \vec{e}_3 \}## to another with a basis ##\{\vec{e}'_1, \vec{e}'_2, \vec{e}'_3 \}##

The transformation equation for the

\vec{e}'_1 \cdot \vec{e}_1 & \vec{e}'_1 \cdot \vec{e}_2 & \vec{e}'_1 \cdot \vec{e}_3 \\

\vec{e}'_2 \cdot \vec{e}_1 & \vec{e}'_2 \cdot \vec{e}_2 & \vec{e}'_2 \cdot \vec{e}_3 \\

\vec{e}'_3 \cdot \vec{e}_1 & \vec{e}'_3 \cdot \vec{e}_2 & \vec{e}'_3 \cdot \vec{e}_3 \end{bmatrix}

\

\begin{pmatrix}v_1\\v_2\\v_3\end{pmatrix}

$$Now since ##\vec{v} = v_i \vec{e}_i = v'_i\vec{e}'_i##, and the matrix is not the identity, the structures ##\begin{pmatrix}v'_1\\v'_2\\v'_3\end{pmatrix}## and ##\begin{pmatrix}v_1\\v_2\\v_3\end{pmatrix}## in the above expression

Similarly, in different contexts we sometimes put vectors in such a column structure, like ##\begin{pmatrix}\vec{e}_1\\\vec{e}_2\\\vec{e}_3\end{pmatrix}## e.g. when looking at how the basis transforms.

It appears to me then that something like ##\begin{pmatrix}a\\b\\c\end{pmatrix}## can represent the raw tuple ##(a,b,c)##, or an expression with certain basis vectors like ##a\vec{e}_1 + b\vec{e}_2 + c\vec{e}_3## , depending on the context. I wondered if someone could clarify whether this is along the right lines?

The transformation equation for the

*components*of ##\vec{v}## looks something like$$\begin{pmatrix}v'_1\\v'_2\\v'_3\end{pmatrix} = \begin{bmatrix}\vec{e}'_1 \cdot \vec{e}_1 & \vec{e}'_1 \cdot \vec{e}_2 & \vec{e}'_1 \cdot \vec{e}_3 \\

\vec{e}'_2 \cdot \vec{e}_1 & \vec{e}'_2 \cdot \vec{e}_2 & \vec{e}'_2 \cdot \vec{e}_3 \\

\vec{e}'_3 \cdot \vec{e}_1 & \vec{e}'_3 \cdot \vec{e}_2 & \vec{e}'_3 \cdot \vec{e}_3 \end{bmatrix}

\

\begin{pmatrix}v_1\\v_2\\v_3\end{pmatrix}

$$Now since ##\vec{v} = v_i \vec{e}_i = v'_i\vec{e}'_i##, and the matrix is not the identity, the structures ##\begin{pmatrix}v'_1\\v'_2\\v'_3\end{pmatrix}## and ##\begin{pmatrix}v_1\\v_2\\v_3\end{pmatrix}## in the above expression

*can't*be ##\vec{v}##. Instead, they just appear to be 3-tuples of numbers with no apparent basis.Similarly, in different contexts we sometimes put vectors in such a column structure, like ##\begin{pmatrix}\vec{e}_1\\\vec{e}_2\\\vec{e}_3\end{pmatrix}## e.g. when looking at how the basis transforms.

It appears to me then that something like ##\begin{pmatrix}a\\b\\c\end{pmatrix}## can represent the raw tuple ##(a,b,c)##, or an expression with certain basis vectors like ##a\vec{e}_1 + b\vec{e}_2 + c\vec{e}_3## , depending on the context. I wondered if someone could clarify whether this is along the right lines?

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