18 questions linked to/from Intuition behind Matrix Multiplication
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### Why, historically, do we multiply matrices as we do?

Multiplication of matrices — taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product — is not a very ...
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### Is “basis times square matrix” a new basis?

Suppose we have a vector space $V = (K, +, \cdot)$. Let $B$ be a basis for $V$. Now we take an arbitrary square matrix $S \neq 0$. $BS$ is just a linear combination of $B$. Thus $BS$ should be a new ...
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### Morphism between matrices and linear equations

I'm currently a beginner at linear algebra. So, in some books I see authors start defining linear equations and then they define matrices and, supposedly, the definition of associative matrix is to ...
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### What does matrix multiplication have to do with scalar multiplication?

Why are matrix and scalar multiplication denoted the same way and treated as the same operation in standard mathematical notation? This is always a source of confusion for me because they have ...
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### show that $((GL(2,\mathbb{R}),\bullet)$ is a group

Let $G(2,\mathbb R)=\{\text{All invertible }2 \times 2\text{ matrices over }\mathbb{R}\}$. Then i want to show that $((G(2,\mathbb{R}),\bullet)$ is a group, where $\bullet$ is multiplication of ...
When I took linear algebra class in my freshman year, the multiplication operation for matrices was defined without any apparent motivation. Given an $m$-times-$n$ matrix $A$ and an $n$-times-$p$ ...