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Emilio Novati
Emilio Novati
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I know it's probably a silly question, but I'm trying to figure out why was matrix multiplication (the standard one) defined the way it was defined.

I know that it was defined like that so we would gain invariance under change of basis: $PA^{-1}+PBP^{-1}=P(A+B)P^{-1}$$PAP^{-1}+PBP^{-1}=P(A+B)P^{-1}$ and $(PAP^{-1})(PBP^{-1})=PABP^{-1}$ and that ofcourse the case.

But another explanation that was suggested is: "We defined matrix multiplication this way so that if $A$ is the matrix of a linear transformation $T_1$ with respect to some basis $s$ and $B$ is the matrix of a linear transformation $T_2$ with respect to the same basis $s$ then $AB$ is the matrix of $T_1$ composition with $T_2$ (I don't know the command for composition operator) with respect to basis $s$.

Again, this is a completely legitimate aspiration, but I fail to see why it follows. Why is linear transformation composition equivalent to matrix multiplication?

I know it's probably a silly question, but I'm trying to figure out why was matrix multiplication (the standard one) defined the way it was defined.

I know that it was defined like that so we would gain invariance under change of basis: $PA^{-1}+PBP^{-1}=P(A+B)P^{-1}$ and $(PAP^{-1})(PBP^{-1})=PABP^{-1}$ and that ofcourse the case.

But another explanation that was suggested is: "We defined matrix multiplication this way so that if $A$ is the matrix of a linear transformation $T_1$ with respect to some basis $s$ and $B$ is the matrix of a linear transformation $T_2$ with respect to the same basis $s$ then $AB$ is the matrix of $T_1$ composition with $T_2$ (I don't know the command for composition operator) with respect to basis $s$.

Again, this is a completely legitimate aspiration, but I fail to see why it follows. Why is linear transformation composition equivalent to matrix multiplication?

I know it's probably a silly question, but I'm trying to figure out why was matrix multiplication (the standard one) defined the way it was defined.

I know that it was defined like that so we would gain invariance under change of basis: $PAP^{-1}+PBP^{-1}=P(A+B)P^{-1}$ and $(PAP^{-1})(PBP^{-1})=PABP^{-1}$ and that ofcourse the case.

But another explanation that was suggested is: "We defined matrix multiplication this way so that if $A$ is the matrix of a linear transformation $T_1$ with respect to some basis $s$ and $B$ is the matrix of a linear transformation $T_2$ with respect to the same basis $s$ then $AB$ is the matrix of $T_1$ composition with $T_2$ (I don't know the command for composition operator) with respect to basis $s$.

Again, this is a completely legitimate aspiration, but I fail to see why it follows. Why is linear transformation composition equivalent to matrix multiplication?

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Oria Gruber
Oria Gruber
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Why does matrix multiplication represent linear transformation compositions?

I know it's probably a silly question, but I'm trying to figure out why was matrix multiplication (the standard one) defined the way it was defined.

I know that it was defined like that so we would gain invariance under change of basis: $PA^{-1}+PBP^{-1}=P(A+B)P^{-1}$ and $(PAP^{-1})(PBP^{-1})=PABP^{-1}$ and that ofcourse the case.

But another explanation that was suggested is: "We defined matrix multiplication this way so that if $A$ is the matrix of a linear transformation $T_1$ with respect to some basis $s$ and $B$ is the matrix of a linear transformation $T_2$ with respect to the same basis $s$ then $AB$ is the matrix of $T_1$ composition with $T_2$ (I don't know the command for composition operator) with respect to basis $s$.

Again, this is a completely legitimate aspiration, but I fail to see why it follows. Why is linear transformation composition equivalent to matrix multiplication?