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03-onlinear - ON SOLUTIONS OF LINEAR EQUATIONS Homework...

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10/1/2004, ON SOLUTIONS OF LINEAR EQUATIONS Math 21b, O. Knill Homework Section 1.3 4,14,34,48,50,26*,46*, due 10/4/2004 MATRIX. A rectangular array of numbers is called a matrix . A = a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n · · · · · · · · · · · · a m 1 a m 2 · · · a mn = w w w w v v v 1 2 3 4 1 2 3 A matrix with m rows and n columns is called a m × n matrix. A matrix with one column is a column vector . The entries of a matrix are denoted a ij , where i is the row number and j is the column number. ROW AND COLUMN PICTURE. Two interpretations A~x = - ~w 1 - - ~w 2 - . . . - ~w m - | ~x | = ~w 1 · ~x ~w 2 · ~x . . . ~w m · ~x A~x = | | · · · | ~v 1 ~v 2 · · · ~v n | | · · · | x 1 x 2 · · · x m = x 1 ~v 1 + x 2 ~v 2 + · · · + x m ~v m = ~ b . Row picture: each b i is the dot product of a row vector ~w i with ~x . Column picture: ~ b is a sum of scaled column vectors ~v j . ”Row and Column at Harvard” EXAMPLE. The system of linear equations 3 x - 4 y - 5 z = 0 - x + 2 y - z = 0 - x - y + 3 z = 9 is equivalent to A~x = ~ b , where A is a coefficient matrix and ~x and ~ b are vectors .
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