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After watching the wonderful playlist of Linear Algebra on the YouTube channel 3blue1brown, I realized that vectors and matrices are, fundamentally, different concepts:

  • Vectors are numerical entities in a $n$-dimensional space.
  • Matrices are how theses numbers are (linearly) transformed.

I am restricting it to a Linear Algebra. Of course, for multilinear algebra, matrices (and tensors) are, in fact, numerical entities in the same way that a vector is. Regardless, for Linear Algebra, this argument seems to have a much deeper interpretation that what is commonly said about vectors and matrices:

A vector is just a one-column matrix

I am not saying that such answer is wrong, once denoting a vector as a column is merely a convention. But doesn't it seem that this argument leaves out the main purpose of using matrices in linear algebra, which is apply linear transformations on vectors though matrix product,i.e., $\mathbf{Ax}$?

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    $\begingroup$ You can think of a vector in $V$ as a linear transformation $\mathbb{R}\to V$. This aligns with both thinking of matrices as linear transformations and vectors as 1 column matrices. I would disagree that vectors and matrices are "fundamentally different" $\endgroup$
    – IsAdisplayName
    Dec 11, 2022 at 23:32
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    $\begingroup$ Adding to what IsAdisplayName said, often in maths the vector space you are working over is a set of matrices (for example in Lie algebra) so there isn't really any clear difference between the two concepts beyond what vector space you are concerned with. $\endgroup$
    – Fishbane
    Dec 11, 2022 at 23:58

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A matrix is a concrete instance of a linear transformation when you've chosen a specific basis. However, when you move from linear algebra over the field of Real Numbers to an abstract field $F$ it becomes more helpful to view linear algebra more abstractly as linear transformations of vectors instead rectangular grids of numbers.

If $V$ is a vector space over ($F$) and $a \in V$, then a linear transformation $T$ is a function from vector space $V$ to some other (possibly identical) vector space $W$ (over $F$) such that $T(ca)=cT(a), T(a+b) = T(a) + T(b)\;\;\forall a,b \in V, c \in F$.

In this viewpoint, the difference between the vector and its transformation becomes clear. However, in the matrix viewpoint, you can have vector space of, say, $2\times 2$ symmetric matrices and a $2\times 1$ column vector being the linear transformation, which will be a transformation from $S_{2,2}$ to $\mathbb R^2$, in this case the roles are flipped.

Note: as pointed out by ex.nihil, if we want to keep the usual format of $Tv$ for a linear transformation $T$ applied to vector $v$ then we’d use the $1 \times 2$ row vector as the transformation $T$ applied to the matrix as $v$.

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  • $\begingroup$ In your last example, I believe you mean that the $2×1$ vector is acting as a right-multiplication transform; is this true? Because otherwise—when using the more typical left-multiplication for transforms, your example should instead be a $1×2$ vector that left-multiplies the matrix to map it from $S_{2,2}$ into $\mathbb{R}^2$. I know this distinction may be pedantic, but I guess it clarifies notational confusion. Please correct me if I'm wrong! $\endgroup$
    – ex.nihil
    Aug 19 at 16:33
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    $\begingroup$ @ex.nihil ah, yes you are correct — wanted to keep to an actual column vector so yes it would be a right transform. As you daod, I’d need to have its transpose to the preserve the form “transform”x”vector” $\endgroup$
    – Annika
    Aug 19 at 16:47
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    $\begingroup$ @ex.nihil I updated my post to reflect this nuance. $\endgroup$
    – Annika
    Aug 19 at 17:57

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