A1_soln

A1_soln - Math 235 1. Find a matrices. 3 2 a) A = 1...

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Math 235 Assignment 1 Solutions 1. Find a basis for the row space, column space and nullspace of each of the following matrices. a) A = 3 6 1 2 4 1 1 2 0 . Solution: Row-reducing A we ﬁnd that its RREF is 1 2 0 0 0 1 0 0 0 . Hence, a basis for the row space is { (1 , 2 , 0) , (0 , 0 , 1) } , a basis for the column space is 3 2 1 , 1 1 0 and a basis for the nullspace is - 2 1 0 . b) B = 0 1 0 - 2 1 2 1 - 1 2 4 3 - 1 . Solution: Row-reducing B we ﬁnd that its RREF is 1 0 0 2 0 1 0 - 2 0 0 1 1 . Hence, a basis for the row space is { (1 , 0 , 0 , 2) , (0 , 1 , 0 , - 2) , (0 , 0 , 1 , 1 , ) } , a basis for the column space is 0 1 2 , 1 2 4 , 0 1 3 , and a basis for the nullspace is - 2 2 - 1 1 . 2. Let A be a 6 × 3 matrix with rank 3. What is dim(nul A ), dim(row A ), rank A T ? Solution: The number of parameters in the general solution of the homogeneous system A~x = ~ 0 is # of columns - rank=3-3=0. Hence the dimension of the nullspace is 0. The dimension of the row space equals the rank of the matrix hence dim(row A ) = 3. The rank of A T equals the rank of A so rank A T = 3.

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2 3. Find a basis for the range and null space of the following linear maps and verify the Rank-Nullity theorem. a)
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This note was uploaded on 10/12/2010 for the course MATH 235 taught by Professor Celmin during the Fall '08 term at Waterloo.

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A1_soln - Math 235 1. Find a matrices. 3 2 a) A = 1...

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