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About an hour ago, I asked this question: What is Matrix?

I was confused (and to a certain extend, still am) about where the properties of matrices come from. My question was marked as a duplicate of this other question: What's a matrix?, which I can certainly understand why.

In the latter question, the repliers explained that matrices are usually used to represent a transformation in vector spaces.

For example, one could represent the set of equations:

$3x+2y+5z $

$9x-4y+z$

$-15x+2z$

as the following matrix:

$$ \begin{pmatrix} 3 & 2 & 5 \\ 9 & -4 & 1 \\ -15 & 0 & 2 \\ \end{pmatrix} \begin{pmatrix} x \\ y\\ z\\ \end{pmatrix} $$

If I understood correctly, the addition rule for matrices, in which we add each corresponding entry of two matrices, could be seen as a way of representing the sum of some sets of linear equations, or, more generally, some sets of vectors.

For instance, if we wanted to add $(3x+2y+5z)+(4x+6y-8z)$, one could write

$$ \begin{pmatrix} 3 & 2 & 5 \\ \end{pmatrix} \begin{pmatrix} x \\ y\\ z\\ \end{pmatrix}+ \begin{pmatrix} 4 & 6 & -8 \\ \end{pmatrix} \begin{pmatrix} x \\ y\\ z\\ \end{pmatrix}= \begin{pmatrix} 7 & 8 & -3 \\ \end{pmatrix} \begin{pmatrix} x \\ y\\ z\\ \end{pmatrix} $$

...right?

Specifically, if we have two matrices, $A$ & $B$, each one with $n$ rows and $m$ columns, the addition rule states that each entry $a_i,_j$ ($i$th row, $j$th column) of $A$ should be added with each entry $b_i,_j$ of $B$, creating a new matrix, $C$, in which each entry $c_i,_j=a_i,_j + b_i,_j$.

Q1. Could this addition rule of matrices be simply seen as a way of adding some sets of vectors?

Q2. If this is the case, why is it that, in order to add two matrices, we need both of these to have the same number of rows and columns? Is it because if we have a different number of rows/columns, then we wouldn't know which vectors correspond with each other?

Any thoughts/ideas would be really appreciated.

Sam
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    You use the word "sets" here which has some deep mathematical significance which I don't think your usage here intends. If you prefer to think of the columns as vectors, sure... that's fine but not necessary. If you do think of it that way, then remember that only vectors from the same space have a well defined addition leading you to needing the same number of rows, and the necessity of having each column having a partner to be added to makes it so they need the same number of columns. – JMoravitz Aug 19 '17 at 06:33
  • Personally, I dislike trying to justify the why's of why matrix addition/multiplication/etc... are defined the way they are, and instead prefer to take the definitions as exactly that, definitions. It so happens that the way matrices and the operations surrounding them are defined happen to be useful and coincide with other mathematical objects, but at the time of their creation they didn't yet have such a deep theory surrounding them that was well known or understood. – JMoravitz Aug 19 '17 at 06:35

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Matrices represent linear transformations. If $f$ and $g$ are linear transformations, then so is $h$ defined by $$h(u)=f(u)+g(u).$$ It is natural to denote $h$ by $f+g$.

If $f$ and $g$ are represented by matrices $A$ and $B$, then $f+g$ is represented by $f+g$.

Angina Seng
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