Let $v_1,...,v_n\in\Bbb R^d$ be a sequence of vectors. When we say that we "linearly transform" this sequence, we mean that we apply a linear transformation $T\in\Bbb R^{d\times d}$ to each vector in the sequence individually. This can also be expressed as applying the transformations $T$ from the left to the following matrix, and then reading off the columns of the resulting matrix:

$$ \newcommand{\verts}{\rule{0.2pt}{1.3em}} \Phi(\boldsymbol v):=\begin{bmatrix} \verts & \!\verts & & \verts \\[-1ex] \,v_1 & \!v_2 & \!\!\!\cdots\!\!\! & v_n\, \\ \verts & \!\verts & & \verts \end{bmatrix}\in\Bbb R^{d\times n} $$

Question: Is there an established name for what I am doing when I apply a linear transformation $S\in\Bbb R^{n\times n}$ from the right to this matrix and then read off the columns?

$$v_1,...,v_n \,\overset S\mapsto\, w_1,...,w_n,\quad \text{if}\,\; \Phi(\boldsymbol w)=\Phi(\boldsymbol v)S.$$

Examples of this are:

• permuting the vectors in the sequence ($S$ is a permutation matrix).
• scaling each vector individually ($S$ is a diagonal matrix).

I suppose these sort of transformations can be called column operations on $\Phi(\boldsymbol v)$, but I wonder specifically whether there is a name for the operations when we think of vector sequences and want to avoid thinking of matrices and explicit basis representations. I am mostly interested in the case when $S$ is invertible.

Let $v_1,...,v_n\in\Bbb R^d$ be a sequence of vectors. When we say that we "linearly transform" this sequence, we mean that we apply a linear transformation $T\in\Bbb R^{d\times d}$ to each vector in the sequence individually. This can also be expressed as applying the transformations $T$ from the left to the following matrix, and then reading off the columns of the resulting matrix:

$$ \newcommand{\verts}{\rule{0.2pt}{1.3em}} \Phi(\boldsymbol v):=\begin{bmatrix} \verts & \!\verts & & \verts \\[-1ex] \,v_1 & \!v_2 & \!\!\!\cdots\!\!\! & v_n\, \\ \verts & \!\verts & & \verts \end{bmatrix}\in\Bbb R^{d\times n} $$

Question: Is there an established name for what I am doing when I apply a linear transformation $S\in\Bbb R^{n\times n}$ from the right to this matrix and then read off the columns?

$$v_1,...,v_n \,\overset S\mapsto\, w_1,...,w_n,\quad \text{if}\,\; \Phi(\boldsymbol w)=\Phi(\boldsymbol v)S.$$

Examples of this are:

• permuting the vectors in the sequence ($S$ is a permutation matrix).
• scaling each vector individually ($S$ is a diagonal matrix).

Neither of this falls under "linearly transforming the sequence" with the usual meaning. But it is still linear in some sense, and I would like to have a suitable terminology to refer to these transformations.

I suppose these sort of transformations can be called column operations on $\Phi(\boldsymbol v)$, but I wonder specifically whether there is a name for the operations when we think of vector sequences and want to avoid thinking of matrices and explicit basis representations. I am mostly interested in the case when $S$ is invertible.

If I am given a sequence $v_1,...,v_n\in\Bbb R^d$ of vectors, applying a linear transformation $T\in\Bbb R^{d\times d}$ to each vector in the sequence can also be expressed as applying the transformations $T$ from the left to the following matrix, and then reading off the columns of the resulting matrix:

$$ \newcommand{\verts}{\rule{0.2pt}{1.3em}} \Phi(\boldsymbol v):=\begin{bmatrix} \verts & \!\verts & & \verts \\[-1ex] \,v_1 & \!v_2 & \!\!\!\cdots\!\!\! & v_n\, \\ \verts & \!\verts & & \verts \end{bmatrix}\in\Bbb R^{d\times n} $$

Question: Is there a name for the sort of operations on a sequence of vectors that can be expressed as applying a linear transformation $S\in\Bbb R^{n\times n}$ from the right to this matrix and then reading off the columns?

$$v_1,...,v_n \,\overset S\mapsto\, w_1,...,w_n,\quad \text{if}\,\; \Phi(\boldsymbol w)=\Phi(\boldsymbol v)S.$$

Examples of this are:

• permuting the vectors in the sequence ($S$ is a permutation matrix).
• scaling each vector individually ($S$ is a diagonal matrix).

I suppose they can be called column operations on $\Phi(\boldsymbol v)$, but I wonder specifically whether there is a name for the operations when we think of vector sequences and want to avoid thinking of matrices and explicit basis representations. I am mostly interested in the case when $S$ is invertible.

Let $v_1,...,v_n\in\Bbb R^d$ be a sequence of vectors. When we say that we "linearly transform" this sequence, we mean that we apply a linear transformation $T\in\Bbb R^{d\times d}$ to each vector in the sequence individually. This can also be expressed as applying the transformations $T$ from the left to the following matrix, and then reading off the columns of the resulting matrix:

$$ \newcommand{\verts}{\rule{0.2pt}{1.3em}} \Phi(\boldsymbol v):=\begin{bmatrix} \verts & \!\verts & & \verts \\[-1ex] \,v_1 & \!v_2 & \!\!\!\cdots\!\!\! & v_n\, \\ \verts & \!\verts & & \verts \end{bmatrix}\in\Bbb R^{d\times n} $$

Question: Is there an established name for what I am doing when I apply a linear transformation $S\in\Bbb R^{n\times n}$ from the right to this matrix and then read off the columns?

$$v_1,...,v_n \,\overset S\mapsto\, w_1,...,w_n,\quad \text{if}\,\; \Phi(\boldsymbol w)=\Phi(\boldsymbol v)S.$$

Examples of this are:

• permuting the vectors in the sequence ($S$ is a permutation matrix).
• scaling each vector individually ($S$ is a diagonal matrix).

I suppose these sort of transformations can be called column operations on $\Phi(\boldsymbol v)$, but I wonder specifically whether there is a name for the operations when we think of vector sequences and want to avoid thinking of matrices and explicit basis representations. I am mostly interested in the case when $S$ is invertible.