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from which we obtain
Thus as guaranteed by Formula 8. We leave it for the reader to verify that 9 also holds.
A Dot Product View of Matrix Multiplication is an matrix and
Dot products provide another way of thinking about matrix multiplication. Recall that if
is an matrix, then the th entry of is
which is the dot product of the ith row vector of A
and the jth column vector of B
Thus, if the row vectors of A are , , …, and the column vectors of B are , , …, , then the matrix product can be
expressed as
(10)
In particular, a linear system can be expressed in dot product form as
(11)
where , , …, are the row vectors of A, and , , …, are the entries of b.
EXAMPLE 6 A Linear System Written in Dot Product Form
The following is an example of a linear system expressed in dot product form 11.
