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I've read this post: Row vector vs. Column vector and while it helped me understand that the differences seems to be dependent on what you're trying to do, I still find myself struggling to tell the difference between a matrix of column vectors and a matrix of row vectors.

If I have a matrix: $$ \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} $$

Is the first matrix two column vectors, \begin{bmatrix} 1 \\ 3 \end{bmatrix} and \begin{bmatrix} 2 \\ 4 \end{bmatrix} ?

Or is it two row vectors [ 1 2 ] and [ 3 4 ]?

I've noticed that in the Coursera course I'm taking when it asks me to do matrix multiplication, the left matrix is row vectors, and the right matrix is column vectors. Is that always the convention or is there some way of knowing?

Leo Holman
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2 Answers2

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The matrix consists of $2$ rows and $2$ columns. We can write the matrix is an element of $\mathbb{R}^{2 \times 2}$.

As for your question regarding matrix multiplication, it is the definition of the matrix multiplication that we define if $C=AB$, where $A\in \mathbb{R}^{m \times p}$ and $B\in \mathbb{R}^{p \times n}$, then $C\in \mathbb{R}^{m \times n}$ where $C_{ij}=\sum_{k=1}^pA_{ik}B_{kj}$, fixing $i$ and $j$, notice that as we increment $k$, we travel along the $i$-th row of $A$ and the $j$-th column of $B$>

Remark: if you want to compute

$$\begin{bmatrix}1 & 2 \\ 3 & 4 \end{bmatrix}\begin{bmatrix}2 \\ 3\end{bmatrix},$$

From the definition, we compute $$\begin{bmatrix} \begin{bmatrix} 1 & 2 \end{bmatrix}\begin{bmatrix}2 \\ 3\end{bmatrix} \\ \begin{bmatrix} 3 & 4 \end{bmatrix}\begin{bmatrix}2 \\ 3\end{bmatrix} \end{bmatrix}$$

but you can also verify that it is equal to $$2\begin{bmatrix} 1 \\ 3\end{bmatrix} + 3\begin{bmatrix} 2 \\ 4\end{bmatrix}.$$

Siong Thye Goh
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  • So I understand what you're saying to mean, the matrix itself is simultaneously two row vectors stacked on top of each other, OR two column vectors next to each other. So when we do matrix multiplication we do rows on the left and columns on the right simply because that's the definition of matrix multiplication, is that right? So we would have to know something more about the problem domain to say something about the significance of the values making more sense as rows or vectors, but the matrix itself can't tell you. Is that right? – Leo Holman Feb 29 '20 at 15:46
  • yes, if you arrange $10$ chocolates in rows of $2$ aligned to each other, you can also described it as $5$ columns as well. – Siong Thye Goh Feb 29 '20 at 15:51
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You can say it is a bunch of column vectors because it shows where $\imath, \jmath$ respectively are taken by the matrix as a linear transformation

Nεo Pλατo
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