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Unformatted text preview: Math 1b — Matrix Multiplication If A has rows a i and B has columns b j , then AB has, by definition, a i b j as the entry in row i and column j . The matrix AB is the matrix of dot products of rows of A with columns of B . Here are some simple properties and facts about matrix multiplication. These rules follow directly from the definition of matrix multiplication. Small examples can help understanding. 1. A (row) vector times a matrix is a linear combination of the rows of that matrix (and the coeﬃcients are the entries of the vector): ( c 1 c 2 ... c ` ) ⎛ ⎜ ⎜ ⎝ — a 1 — — a 2 — . . . — a ` — ⎞ ⎟ ⎟ ⎠ = c 1 a 1 + c 2 a 2 + ... + c ` a ` . 2. The rows of the matrix product AB are (rows of A ) times B . ⎛ ⎜ ⎜ ⎝ — a 1 — — a 2 — . . . — a ` — ⎞ ⎟ ⎟ ⎠ B = ⎛ ⎜ ⎜ ⎝ — a 1 B — — a 2 B — . . . — a ` B — ⎞ ⎟ ⎟ ⎠ ....
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 Winter '08
 Aschbacher
 Linear Algebra, Multiplication, Vector Space, Dot Product, Ring

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