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.
–
Calvin Lin
Staff
·
4 years, 6 months ago

it is just a row space by column space multiplication and summation of all such possible multiplications .......
–
Jaydutt Kulkarni
·
4 years, 6 months ago

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TopNewestInstead 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. – Calvin Lin Staff · 4 years, 6 months ago

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Found this. – Zi Song Yeoh · 4 years, 6 months ago

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it is just a row space by column space multiplication and summation of all such possible multiplications ....... – Jaydutt Kulkarni · 4 years, 6 months ago

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