# Matrices

Could anyone please explain why matrix multiplication is defined in such a weird way?

Note by Pranav Chakravarthy
5 years, 5 months ago

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Instead of being weird, it's actually very natural. Matrices are meant to represent linear maps, and using that information would help you understand why matrix multiplication (and addition) is defined as such.

For example, if I have 2 linear maps $$f: \mathbb{R}^3 \rightarrow \mathbb{R}^2, g: \mathbb{R}^2 \rightarrow \mathbb{R}^3$$ given by $$f( a, b, c) = (11a + 12b +13c, 21a + 22b + 23c)$$ and $$g( x, y) = (99x + 98 y, 89x + 88 y, 79x + 78 y)$$, what is the value of $$g (f (a, b, c))$$ and $$f(g(x, y) )$$? If you do not expand the terms [Treat $$11$$ as $$a_{11}$$], you will find that the composition of these linear maps results in matrix multiplication.

Repeat the above for the appropriate scenario to relate the addition of 2 linear maps $$(f+g) (x) = f(x) + g(x)$$ to matrix addition.

Staff - 5 years, 5 months ago

Found this.

- 5 years, 5 months ago

it is just a row space by column space multiplication and summation of all such possible multiplications .......

- 5 years, 5 months ago