lec1_3

# lec1_3 - NOTES ON SECTION 1.3 MA265 Key words Dot product...

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NOTES ON SECTION 1 . 3 MA265, SECTIONS 41, 52 Key words Dot product Coefficient matrix inner dimensions Matrix product augmented matrix Given two n -vectors (they have to be the same size), the dot product is defined as the obvious general- ization of this example: [1 , - 4 , 2 , π ] · 1 2 1 2 = 1 * (1) + ( - 4) * 2 + 2 * 1 + π * 2 = - 5 + 2 π Given two matrices A = a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n a 31 a 32 . . . a 3 n . . . . . . . . . a m 1 a m 2 . . . a mn , b = b 11 b 12 . . . b 1 p b 21 b 22 . . . b 2 p b 31 b 32 . . . b 3 p . . . . . . . . . b n 1 b m 2 . . . b np the matrix product A * B is the matrix with i, j -entry equal to n X k =1 a i,k * b k,j i.e. it is the “dot product” 1 of the i th row of A and the j th column of B . It only makes sense if the inner dimensions are equal: A = m × n and B = n × p , and the product makes sense because “ n = n ”. You can’t multiply, e.g., a 3 × 2 with a 3 × 2 matrix.

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