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Unformatted text preview: Chapter 3 SSM: Linear Algebra Chapter 3 3.1 1. Find all ~x such that A~x = ~ 0: " 1 2 . . . 3 4 . . . #→ " 1 . . . 1 . . . # , so that x 1 = x 2 = 0. ker( A ) = { ~ } . 3. Find all ~x such that A~x = ~ 0; note that all ~x in R 2 satisfy the equation, so that ker( A ) = R 2 = span( ~ e 1 , ~ e 2 ). 5. Find all ~x such that A~x = ~ 0. 1 1 1 . . . 1 2 3 . . . 1 3 5 . . . → 1 1 . . . 1 2 . . . . . . ; x 1 = x 3 x 2 = 2 x 3 ; x 1 x 2 x 3 = t 2 t t ker( A ) = span 1 2 1 . 7. Find all ~x such that A~x = ~ 0. Since rref( A ) = I 3 we have ker( A ) = { ~ } . 9. Find all ~x such that A~x = ~ 0. Solving this system yields ker( A ) = { ~ } . 11. Solving the system A~x = ~ 0 we find that ker( A ) = span  2 3 1 . 13. Solving the system A~x = ~ 0 we find that ker( A ) = span  2 1 ,  3 2 1 1 , 1 . 15. By Fact 3.1.3, the image of A is the span of the columns of A : im( A ) = span 1 1 , 1 2 , 1 3 , 1 4 . 64 SSM: Linear Algebra Section 3.1 Since any two of these vectors span all of R 2 already, we can write im( A ) = span 1 1 , 1 2 . 17. By Fact 3.1.3, im( A ) = span 1 3 , 2 4 = R 2 (the whole plane). 19. Since the four column vectors of A are parallel, we have im( A ) = span 1 2 , a line in R 2 . 21. By Fact 3.1.3, im( A ) = span 4 1 5 , 7 9 6 , 3 2 8 . It is hard to tell by inspection whether this span is a plane in R 3 or all of R 3 ; we need to find out whether the third column vector is a linear combination of the first two. 4 7 . . . 3 1 9 . . . 2 5 6 . . . 8 → 1 . . . 1 . . . . . . 1 This shows that the third column vector is not contained in the span of the first two, so that im( A ) = R 3 . 23. im( T ) = R 2 and ker( T ) = { ~ } , since T is invertible (see Summary 3.1.8). 25. im( T ) = R 2 and ker( T ) = { ~ } , since T is invertible (see Summary 3.1.8). 27. Let f ( x ) = x 3 x = x ( x 2 1) = x ( x 1)( x + 1). Then im( f ) = R , since lim x →∞ f ( x ) = ∞ and lim x →∞ f ( x ) =∞ but the function fails to be invertible since the equation f ( x ) = 0 has three solutions, x = 0, 1, and 1. 29. Use spherical coordinates (see any good text on multivariable calculus): f φ θ = sin( φ ) cos( θ ) sin( φ ) sin( θ ) cos( φ ) 65 Chapter 3 SSM: Linear Algebra 31. The plane x + 3 y + 2 z = 0 is spanned by the two vectors  2 1 and  3 1 , for example....
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This homework help was uploaded on 01/23/2008 for the course MATH 253B taught by Professor Bretsch during the Spring '08 term at Colby.
 Spring '08
 BRETSCH
 Linear Algebra, Algebra

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