20105ee205A_1_2010ee205A_1_HW2_sol

20105ee205A_1_2010ee205A_1_HW2_sol - Chapter 2 An...

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Chapter 2 An Introduction to Vector Spaces 1. Suppose { v 1 , . . . , v k } is a linearly dependent set. Then show that one of the vectors must be a linear combination of the others. Answer 2.1 Since { v 1 , . . . , v k } is a linearly dependent set of vectors, there exist k scalars α 1 , . . . , α k , not all zero, such that α 1 v 1 + ··· + α k v k =0 . By reindexing the vectors and scalars if necessary, we can ensure that α 1 ± . Then v 1 = - 1 α 1 ( α 2 v 2 + + α k v k )= γ 2 v 2 + + γ k v k , where γ i = - α i α 1 for i ∈ { 2 , . . . , k } . 2. Let x 1 ,x 2 , . . . , x k IR n be nonzero mutually orthogonal vectors. Show that { x 1 , . . . , x k } must be a linearly independent set. Answer 2.2 This problem is equivalent to showing that X T X is non- singular, where X =( x 1 , . . . , x k ) n × k . But X T X x 1 , . . . , x k ) T ( x 1 , . . . , x k ( x T i x j ) i,j k , which is a diagonal k × k matrix whose diagonal entries x T i x i are nonzero since each x i is nonzero. Thus X T X is nonsingular. 3. Let v 1 , . . . , v n be orthonormal vectors in IR n . Show that Av 1 , . . . , Av n are also orthonormal if and only if A n × n is orthogonal. Answer 2.3 DeFne V v 1 , . . . , v n ) n × n . The matrix V is square with orthonormal columns, so the rank of V is n . Thus V has an 5
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6 CHAPTER 2. AN INTRODUCTION TO VECTOR SPACES inverse, namely V - 1 = V T since V T V = I . Hence VV T = I = V T V , i.e., V is orthogonal. We must show that ( Av i ) T ( Av j )= δ ij , i, j n if and only if A is orthogonal. But this is equivalent to showing that ( AV ) T ( AV I if and only if A is orthogonal. If A is orthogonal, then ( AV ) T ( AV V T ( A T A ) V = I . Conversely, if ( AV ) T ( AV I , then A T A =( T ) A T A ( T V ( AV ) T ( AV ) V T = V IV T = I. But A has an inverse since det A = ± det I ² =0 , so A - 1 = A T . Hence AA T = I = A T A , i.e., A is orthogonal.
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This note was uploaded on 04/18/2011 for the course EE 205A taught by Professor Laub during the Fall '10 term at UCLA.

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20105ee205A_1_2010ee205A_1_HW2_sol - Chapter 2 An...

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