Matrix Multiplication part 2
October 1, 2011 Leave a comment
This is part two of the “boring stuff.” I put that in quotes to evoke a sense of perspective. How boring can it be? We’re still doing math.
Suppose I have linear transformations , and . Let , , and be bases for , , and respectively. What happens when I compose and ? I sure hope that it turns out to me multiplying the matrices and .
Let , , .
Let . I want to know what the entry in the matrix is. So I want to find out what does to a the basis vector and then look at the component of . That will give me the appropriate coefficient.
Of course, we only care about the coefficient of , so we can disregard any time . This gives us:
And of course, this is exactly what you would get if you multiplied matrices the way you were taught.
Okay. I’m pretty sure we’re done with annoying computations for a while.