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pset4sol

# pset4sol - 18.06 Problem Set 4 Solutions Problem 1 Do...

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18.06 Problem Set 4 Solutions Problem 1: Do problem 13 from section 3.6. Solution (a) If m = n then the row space of A equals the column space. FALSE . Counterexample: A = 1 2 3 6 . Here, m = n = 2 but the row space of A contains multiples of (1 , 2) while the column space of A contains multiples of (1 , 3). (b) The matrices A and - A share the same four subspaces. TRUE . The nullspaces are identical because A x = 0 ⇐⇒ ( - A ) x = 0 . The column spaces are identical because any vector v that can be expressed as v = A x for some x can also be expressed as v = ( - A )( - x ). A similar reasoning holds for the two remaining subspaces. (c) If A and B share the same four subspaces then A is a multiple of B . FALSE . Any invertible 2x2 matrix will have R 2 as its column space and row space and the zero vector as its (left and right) nullspace. However, it is easy to produce two invertible 2x2 matrices that are not multiples of each other: A = 1 0 0 2 and B = 2 0 0 1 . Problem 2: Do problem 25 from section 3.6. Solution (a) A and A T have the same number of pivots. TRUE . The number of pivots of A is its column rank, r . We know that the column rank of A equals the row rank of A , which is the column rank of A T . Hence, A T must have the same number of pivots as A . 1

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(b) A and A T have the same left nullspace. FALSE . Counterexample: Take any a 1x2 matrix, such as A = 1 1 . The left nullspace of A contains vectors in R while the left nullspace of A T , which is the right nullspace of A , contains vectors in R 2 , so they cannot be the same.
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pset4sol - 18.06 Problem Set 4 Solutions Problem 1 Do...

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