## How does a matrix times a vector yield a vector?

The title pretty much sums up my quandary. I'm confused as to how a vector "x" with the same number of elements as columns in matrix "A" could yield a vector "b" when multiplied together. I mean, what's stopping "b" from being a matrix instead?
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 This is just how matrix-vector mulitplication is defined. If you're familiar with dot products and such, perhaps this explanation will give you a better understanding of what's going on. Consider the matrix, $$A = \begin{pmatrix} 3 & 4 & 2 \\ -1 & 7 & 6 \\ 9 & -5 & -8\end{pmatrix}$$ This describes a linear operator which can be expressed in terms of dot products and basis vectors. Namely, for any vector $a$, $$\underline A(a) = [(3e_1 + 4 e_2 + 2 e_3) \cdot a]e_1 + [(-e_1 + 7 e_2 + 6 e_3) \cdot a]e_2 + [(9e_1 - 5 e_2 - 8 e_3) \cdot a]e_3$$ Matrix-vector multiplication has been defined in such a way to make this series of dot products easy to quickly evaluate and to make the components of all these various vectors easy to write down in a small space. Nevertheless, this is an entirely equivalent description of what's happening. It's not really "multiplication" at all. It's a linear operator--it's more like a function.

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