Linear Algebra: Homework 7

Transcription

1 Linear Algebra: Homework 7 Alvin Lin August 6 - December 6 Section 3.5 Exercise x Let S be the collection of vectors in R y that satisfy the given property. In each case, either prove that S forms a subspace of R or give a counterexample to show that it does not. Since S R, S is a subspace of R. x y y y x span () Exercise 3 x Let S be the collection of vectors in R y that satisfy the given property. In each case, either prove that S forms a subspace of R or give a counterexample to show that it does not. Since S R, S is a subspace of R. [ x x y x y x ] x span () Exercise 5 x Let S be the collection of vectors y in R 3 that satisfy the given property. In each case, either prove z that S forms a subspace of R or give a counterexample to show that it does not. x y z x x S y x x span z x Since R 3, S is a subspace of R 3.

2 Exercise 7 x Let S be the collection of vectors y in R 3 that satisfy the given property. In each case, either prove z that S forms a subspace of R or give a counterexample to show that it does not. x y + z + Since the zero vector is not in S, S is not a subspace of R 3. Exercise 9 Prove that every line through the origin in R 3 is a subspace of R 3. x a a y t a span a z a 3 a 3 Every line through the origin can be described as the span of a vector through the origin is a subspace of R 3. a a R 3. Therefore, every line a 3 Exercise Suppose S consists of all points in R that are on the x-axis or the y-axis (or both). Is S a subspace of R? Why or why not? { } { } x x R y R y + S S S The subspace S is not closed under addition. Exercise 5 If A is the matrix in Exercise, is v 3 in null(a)? A [ A v 3 ] [ ] / S v / null(a)

3 Exercise 6 7 If A is the matrix in Exercise, is v in null(a)? Exercise 7 3 A A v 4 v null(a) Give bases for row(a), col(a), and null(a). A basis for row(a) {[ ], [ ]} { } basis for col(a), x A x x 3 x x 3 x + x 3 x null(a) x x x basis for null(a) 3

4 Exercise 9 Give bases for row(a), col(a), and null(a). A basis for row(a) {[ ], [ ], [ ]} basis for col(a),, Exercise 7 A x x + x 3 x x 3 x 4 x null(a) x x basis for null(a) Find a basis for the span of the given vectors. 4

5 basis, Exercise 9 Find a basis for the span of the given vectors. [ 3 ] [ ] [ 4 4 ] Exercise basis {e, e, e 3 } Give the rank and nullity of the matrix in Exercise 7. A Exercise 37 rank(a) nullity(a) 3 rank(a) Give the rank and nullity of the matrix in Exercise 9. A rank(a) 3 nullity(a) 4 rank(a) 5

6 Exercise 39 If A is a 3 5 matrix, explain why the columns of A must be linearly dependent. rank(a) min(m, n) A x 5 rank(a) + nullity(a) rank(a) 3 The nullity of A must be at least, therefore A x has a non-trivial solution. Thus, the columns of A are linearly dependent. Since there are 5 columns and only 3 can be linearly independent, they must be linearly dependent. Exercise 4 If A is a 4 matrix, explain why the rows of A must be linearly dependent. There are four rows in total, but only two rows can be linearly independent in a 4 matrix, therefore the rows must be linearly dependent. Exercise 4 If A is a 3 5 matrix, what are the possible values of nullity(a)? rank(a) 3 n rank(a) + nullity(a) 5 rank(a) + nullity(a) 5 nullity(a) 3 nullity(a) nullity(a) nullity(a) 5 Exercise 4 If A is a 4 matrix, what are the possible values of nullity(a)? rank(a) n rank(a) + nullity(a) rank(a) + nullity(a) nullity(a) nullity(a) nullity(a) nullity(a) 6

7 Exercise 43 Find all possible values of rank(a) as a varies. a A 4a a a 4a a a a a 4a a a a a 4a a a a a a + + a a + a a a + + a a a a + a a (a )(a + ) a a, 4 a if a rank(a) if a a 3 otherwise 7

8 Exercise 44 Find all possible values of rank(a) as a varies. a A 3 3 a Exercise 45 Do,, form a basis for R 3? a a + a a a a + a 3 a + ( + a) a a + a + a a 3 3a 4 a a a + a + 3 3a 4 a 3 { if a rank(a) 3 otherwise Since the vectors are linearly independent and form the standard basis vectors, they form a basis for R 3. 8

9 Exercise 47 Do,,, for a basis for R4? 3 Since the vectors form the standard basis vectors, they form a basis for R 4. Exercise 5 Show that w is in span(b) and find the coordinate vector [w] B. B, w 6 c + c [w] B 9

10 Exercise 5 Show that w is in span(b) and find the coordinate vector [w] B. 3 5 B, w c c [w] B 4 Exercise 57 If A is m n, prove that every vector in null(a) is orthogonal to every vector in row(a). A x x R m m y c i a i u row(a) i c [ col (A)... col m (A) ]. c A T. c m x T y x T A T c x T y (Ax) T c x y c c m

11 Exercise 58 If A and B are n n of rank n, prove that AB has rank n. nullity(a) nullity(b) B x A(B x) (AB) x nullity(ab) rank(ab) n nullity(ab) n Exercise 59a Prove that rank(ab) rank(b). n rank(ab) + nullity(ab) rank(b) + nullity(b) nullity(b) dim(null(b)) dim(null(ab)) nullity(ab) rank(ab) + nullity(ab) rank(b) + nullity(b) rank(ab) rank(b) Exercise 6a Prove that rank(ab) rank(a). AB A [ col (B)... col n (B) ] col(ab) col(b) dim(col(ab)) dim(col(b)) rank(ab) rank(b) Exercise 6 Prove that if U is invertible, then rank(ua) rank(a). A IA A U UA rank(u) n rank(a) n rank(u) Prove that if V is invertible, then rank(av ) rank(a). A AI A AV V rank(u) n rank(a) n rank(v )

12 Exercise 6 Prove that an m n matrix A has rank if and only if A can be written as the outer product uv T of a vector u R m and v R n. Suppose: Exercise 63 a A. a n c c n u. w v T A uv T c. rank(a) rank(uv T ) rank(u) rank(a) Prove that an m n matrix A has rank r, prove that A can be written as the sum of r matrices, each of which has rank. rank(a) r dim(row(a)) a A. a m c n w c v + c v + + c r v r c v + c v + + c r v r. c m v + v m v + + c mr v r c c. c m v + + c r c r. c mr v r Because c c. c m, has rank, each of the r matrices has rank.

13 Exercise 64 Prove that, for m n matrices A and B, rank(a + B) rank(a) + rank(b). row i (A + B) row i (A) + row i (B) The rows of A + B can be expressed as linear combinations of the respective rows of A and B. Exercise 65 Let A be an n n matrix such that A. Prove that rank(a) n. A A [ col (A)... col n (A) ] A x col(a) null(a) rank(a) + nullity(a) n rank(a) + rank(a) n rank(a) n rank(a) n Exercise 66 Let A be a skew-symmetric n n matrix. Prove that x T Ax for all x R n. x T Ax (x T Ax) T ) (Ax) T (x T ) T x T A T x x T ( A)x x T Ax x T Ax x T Ax Prove that I + A is invertible. If this is true (I + A)x has only the trivial solution. (I + A)x x + Ax x T x + x T Ax (x T ) x T x + x 3

14 Section 3.6 Exercise Let T A : R R be the matrix transformation corresponding to A 3 where u and v. T A ( u) 8 T A ( v). Find T 3 4 A ( u) and T A ( v), Exercise 3 Prove that the given transformation is a linear transformation, using the definition. x x + y T y x y () ux + v T (u + v) T x u y + v y ux + v x + u y + v y u x + v x u y v y (ux + u y ) + (v x + v y ) (u x u y ) + (v x v y ) T (u) + T (v) cux + cu T (cu) y cu x cu y ux + u c y u x u y ct (u) 4

15 Exercise 5 Prove that the given transformation is a linear transformation, using the definition. x T y x y + z x + y 3z z () ux + v T (u + v) T x u y + v y (ux + v x ) (u y + v y ) + (u z + v z ) (u x + v x ) + (u y + v y ) 3(u z + v z ) (ux u y + u z ) + (v x v y + v z ) (u x + u y 3u z ) + (v x + v y 3v z ) Exercise 7 T (u) + T (v) cux cu T (cu) y + cu z cu x + cu y c3u z ux u c y + u z u x + u y 3u z ct (u) Give a counterexample to show that the given transformation is not a linear transformation. x y T y x Exercise 9 T ( ) () 3 T Give a counterexample to show that the given transformation is not a linear transformation. x xy T y x + y ( ) 3 36 T 3 () 3 T 3 8 5

16 Exercise Find the standard matrix of the linear transformation. x x + y T y x y Exercise 3 Find the standard matrix of the linear transformation. x T y x y + z x + y 3z z 3 Exercise 4 Use matrices to prove the given statements about the transformations from R to R. If R θ denotes a rotation (about the origin) through the angle θ, then R α R β R α + R β. cos α sin α R α sin α cos α cos β sin β R β sin β cos β cos α sin α cos β sin β R α R β sin α cos α sin β cos β cos α cos β sin α sin β cos α sin β sin α cos β sin α cos β + cos α sin β sin α sin β + cos α cos β cos α cos β sin α sin β (cos α sin β + sin α cos β) sin α cos β + cos α sin β cos α cos β sin α sin β cos(α + β) sin(α + β) sin(α + β) cos(α + β) Exercise 4 R α+β (a) If P is a projection, then P P P. P P P P d x d x+d y d xd y d x +d y d xd y d x+d y d y d x +d y d 4 x +d x d y d 3 x dy+dxd3 y (d x+d y) (d x+d y) d 3 xd y+d xd 3 y d 4 y+d xd y (d x +d y ) (d +d ) 6

17 Exercise 44 Let T be a linear transformation from R to R. Prove that T maps a straight line to a straight line or a point. l x + t d T ( x + t d) T ( x) + T (t d) T ( x) + tt ( d) When t, the result is a point, otherwise, the resulting mapping is a line. Exercise 5 Prove that P l (c v) cp l ( v) for an scalar c. ( ) d (c v) P l (c v) d d d ( c( d ) v) d d d ( ) d v c d d d Exercise 53 cp l ( v) Prove that T : R n R m is a linear transformation if and only if: Exercise 54 T (c v + c v ) c T ( v ) + c T ( v ) T (c v + c v ) T (c v ) + T (c v ) c T ( v ) + c T ( v ) Prove that (as noted at the beginning of this section) the range of a linear transformation T : R n R m is the column space of its matrix [T ]. a... a n [T ]... a m... a mn u u. u n T ( u) u a. a m + + u n The range of T is a linear combination of the columns, so it is a subset of the column space. The converse is also true, the column space is a subset of the range. Therefore, the two sets are equal. If you have any questions, comments, or concerns, please contact me at alvin@omgimanerd.tech a n. a mn 7