Matrix multiplication

• When multiplying matrices and , you can write matrix to the top and right of matrix , so that their corners are touching. The resultant matrix will fit in the space below matrix
  • The first entry of the product is given by multiplying, one by one, the entries of the first row of by those of the first column of and adding these products together
  • The second entry of is derived from doing the same thing, but with the first row of with the second column of
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  • Matrix multiplication is not commutative, nor is able to computed for every pair of matrices. is not able to be computed.
• If is a matrix, and is a matrix, then is a matrix.

Definition: Matrix multiplication. If is an matrix whose th entry is is , and is an matrix whose th entry is , then is the matrix with entries:

Properties of matrix multiplication

• Matrix multiplication is associative, meaning the order the operation is applied in is irrelevant
• Matrix multiplication is non-commutative, however. might not be the same as , or might not even be able to be computed (the number of columns and rows do not match)

Multiplication by standard basis vectors

• When multiplying a matrix by the standard basis vector selects out the th column of
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• When you multiply and , and select then th column, it has the same effect as multiplying with the th column of
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• Say that
  • We know that
  • What is the th column of ?
• The th column of can be written as ,

Basis and matrix multiplication

• Set some . Let .
  • We can write this using standard basis vectors as such:
  • We can also represent this as
• This kind of representation will be helpful for linear transformations and change of basis