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Say I have a matrix,

$$\left( \begin{array}{cc} a & d \\ b & e \\ c & f \end{array}\right)$$

I am confused when people say columns, do they mean only the $$\begin{array}{c} a \\ b \\ c \end{array} $$ in the matrix or this vector $$\left( \begin{array}{c} a \\ b \\ c \end{array} \right)$$ like do they mean the entries of column in the matrix or a column matrix with same entries as that in column of the given matrix.

Some lecturers seem to use it interchangeably. I'm not sure if I'm even making sense but for instance, $AX=b$ is the system of equations, where X is a column vector then $AX$ is the linear combinations of columns of A? Wouldn't it be more correct to say, column matrices instead? Or is column a matrix in its own right? Well that can't be right because then I have more questions ...

William
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    Short answer: a "column" means all the things you've mentioned. It doesn't create any problems usually, though. Do you have a specific example in mind where different meanings of "column" lead to substantially different interpretations? – lisyarus Sep 09 '19 at 11:30
  • @lisyarus well if there is no difference between a column and a column matrix then for starters $$\left( \begin{array}{cc} a & d \ b & e \ c & f \end{array}\right) ≠ \left( \begin{array}{cc} {\left(\begin{array}{c}a \ b \ c \end{array}\right) \left(\begin{array}{c}d\ e\ f \end{array}\right)} \end{array}\right)$$ because on the LHS I have a matrix of 3x2 and on the RHS I have a matrix of order 1x2 and there are many more notation problems... – William Sep 09 '19 at 11:36

1 Answers1

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A column matrix is an ordered list of numbers written in a column. However, you are right that column and column matrix are terms which are used interchangeably.

In practice, what is meant will be clear from the context. So, in your example, your lecturer clearly meant that $AX$ is a linear combination of the column matrices $\left( \begin{array}{c} a \\ b \\ c \end{array} \right)$ and $\left( \begin{array}{c} d \\ e \\ f \end{array} \right)$.

Ideally, you should use the terms precisely but don't expect everyone else to do the same!

  • I have a feeling there's something more, because I've read somewhere that a matrix is a vector of vectors so that must mean that column (or row) in the matrix is same as column (or row) vector. But then see my reply below the question. – William Sep 09 '19 at 11:43
  • I would say that a matrix is a rectangular array of numbers. Each of its columns can be thought of as a vector (rather than being a vector). –  Sep 09 '19 at 11:50
  • yes, I agree, it can be "thought of" in that way, but my question is... is it really? – William Sep 09 '19 at 11:52
  • I don't think it becomes a vector until I have put brackets round it, until then its just a column of numbers. –  Sep 09 '19 at 11:54
  • but but... $$ \begin{array}{c} a \ b \ c \end{array} $$ is also a perfectly fine "rectangular array of numbers" so nothing is really stopping us from calling it a matrix. But if you call it a matrix then we're back to square 1, you see my problem? – William Sep 09 '19 at 12:02
  • Suppose we actually say a matrix is a rectangular array of numbers enclosed in brackets. Does that remove the problem? –  Sep 09 '19 at 12:07
  • that was my first thought too but here now I am more confused than ever https://math.stackexchange.com/q/3341975/552998 – William Sep 09 '19 at 12:09
  • Let us continue this discussion in chat. –  Sep 09 '19 at 12:12
  • not sure what's the difference between chatting here and there, though I'm not sure why isn't this question getting attention, could be because of poor formatting? – William Sep 10 '19 at 10:14