Matrix Arithmetic If you don’t remember how to add matrices, you should look it up now. Here we are going to talk about matrix products. Let A ∈ R m × n and B ∈ R n × p . Let’s also say that the matrix A is the coordinate representation of a linear transformation A : R n → R m , and likewise for B and B : R p → R n . Then linear transformation C = AB : R p → R n → R m has as its coordinate representation the matrix C = AB . While it is true that a matrix is a rectangular array of numbers , it will be useful for us to remember that a matrix represents a linear function from one vector space to another: a matrix is a linear transformation . This is precisely why the natural product of two matrices isn’t entrywise, like addition, but instead has the (not as intuitive) form C = [ c ij ] , where c ij = n X k =1 a ik b kj . Let’s let a t i be the i th row of A , and b j be the j th column of B . Then c ij = a t i b j . If we now let c j be the j th column of C , we can write
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This note was uploaded on 12/18/2010 for the course PHYS 5073 taught by Professor Mark during the Fall '10 term at Arkansas.