REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS

Transcription

1 REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS The problems listed below are intended as review problems to do before the final They are organied in groups according to sections in my notes, but it is not forbidden to use techniques from later sections sometimes dramatically easier to solve earlier problems I emphatically do not imply that my solutions are unique or even the best solutions to these problems I hope, at least, that there are no erroneous solutions If you find an error or typo, or have a better solution, be sure to let me know! i Linear Equations and R n a, b and c Solve: x y and x y 7 and x 4 y 7 Solutions: a We interpret this to mean x+ 7 7 and y Rewriting this parametrically set t gives x 7 7 y + t t is any real number b c 4 7 No solution 7 4 x y Date: March 4, 6

2 REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS ii Parametric and Point-Normal Equations for Lines and Planes a Find parametric and point-normal forms for a line in R through the point 6, 4 and perpendicular to the vector, Parametric: Qt 6, 4 + t, Point Normal: x, y 6, 4, b Find parametric and point-normal forms for a plane containing the three points 6, 4, 7,,, and,, V 6, 4, 7,, 6, 4, 6 and W,,,,,, lie in the plane, head and tail V W 4,, 4 Parametric: Qs, t,, + s 6, 4, 6 + t,, Point Normal: x, y,,, 4,, 4 iii Linear Transformations from R n to R m a, b and c Which of the following are linear? Justify your conclusion g x y 4 h x y xy f x y x x + y x + y x + y Solutions: a Not linear, because g b Not linear If linear, it would correspond to the matrix he he he However x y x + y xy x + y c Is linear: fe fe fe x y iv Eigenvalues x y x x + y a, b and c Find real eigenvalues and associated eigenvectors if any for: and 4 5 Solutions: λ a det λ λ + which has no real solutions No real eigenvalues λ + b det λ + λ Real eigenvalues: and λ λ has eigenvector, λ has eigenvector,

3 c REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS λ det λ λ 5 Real eigenvalues: and 5 4 λ 5 λ has eigenvector, λ 5 has eigenvector, v General Vector Spaces and Subspaces a Is the set of by matrices whose determinant is a vector space with the usual matrix addition and scalar multiplication? No: not closed under addition + b Is the set of ordered pairs of integers a vector space with the usual operations? No: not closed under scalar multiplication,, c Prove that the eigenspace for a given eigenvalue of a square matrix is a vector space Suppose that v and w are eigenvectors for eigenvalue λ for matrix M and c is any constant Mv + cw Mv + cmw λv + cλw λv + cw So v + cw is also an eigenvector for this eigenvalue d Is a plane in three dimensional space which contains the origin a subspace of R? What if it doesn t contain the origin? Yes: the equation of a plane through the origin can be written in normal form n x Dot product is linear, so as in the last example the vectors in the plane are closed under scalar multiplication and vector addition The points in any plane not containing cannot be closed under scalar multiplication and so is not a vector space e Is the graph of y x a subspace of R? No:, is in this set but, is not f Is the set consisting of the x and y axes combined a vector subspace of R? No:, and, are in this set but, +, is not

4 4 REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS vi Basis for a Vector Space a Find a basis for the vector space of by matrices with trace Trace is a linear function on matrices so its nullspace is a vector space Obviously not all by matrices have trace ero, so the dimension of the nullspace is at most three Here is a collection of three independent matrices with trace, which must therefore constitute a basis: and and b Find a basis for the plane given by, 6, x, y, { 6,,,,, 6 } These are two independent vectors in the plane, which cannot have dimension exceeding two since it is not all of R So they form a basis c Find a basis for the hyperplane given by, 6,, 5 x, y,, w { 6,,,,,, 6,,,, 5, } These are three independent vectors in the hyperplane, which cannot have dimension exceeding three since it is not all of R 4 So these vectors form a basis d Find a basis for the set of polynomials in the variable t hint: This will be an infinite basis The easiest basis is {, t, t, } where we will not dwell upon the meaning of You know what I mean, right? It is obvious that this set spans the vector space of polynomials, since any polynomial is by definition a linear combination of these powers We need to show it is an independent set Suppose there is a sum a n t n + a n t n + + a of distinct powers of t, and where a n is nonero, which adds to Then dividing both sides by a n t n we find that + a n a n t + + a a n t n But the limit, as t, of the right is, since every term but the first has t in the denominator Since we conclude that there is no such nontrivial linear combination, and the set of non-negative powers of t is independent vii Linear Functions Between Vector Spaces Let W denote the set of polynomials in x of degree 5 or less Consider the linear function d dx : W W a What is the nullspace of d dx? Find a basis d By the mean value theorem applied twice if dx f then fx must be of the form ax + b So {, x } is a basis of the nullspace

5 REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS 5 b What is the image of d dx? Find a basis d dx reduces the degree of a polynomial by exactly if it started at degree 5, 4, d or So the image of dx will be certain polynomials of degree or less On the other hand, d a dx x5 + a x4 + a 6 x + a x a x + a x + a x + a So any third degree polynomial is in the image of this differential operator So {, x, x, x } is a basis for the image c Pick a basis A for W and find a matrix M A,A for d dx How about the easiest? Let A {, x, x, x, x 4, x 5 }, in that order M A,A [ ] [ ] d d dx A dx x A [ ] 6 d dx x5 A viii Nullspace, Columnspace and Solutions a, b and c Find the nullspace and the columnspace for each matrix: and and Solutions: a Nullspace: Span {, 9, 7 } Columnspace: R b Nullspace: Span {,,,,, } Columnspace: Span {, } c Nullspace: {, } Columnspace: R d, e and f Represent the solution to the following equations in parametric form if there is more than one solution using your work from above: x y and x y 7 x and 7 4 y 7 Solutions: d Solution Set:,, + t, 9, 7 for any real t This could also be written as,, + Span {, 9, 7 } e Solution Set: 7,, + s,, + t,, for any real s or t f, only

6 6 REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS ix More on Nullspace and Columnspace a Find the nullspace and the columnspace of Nullspace: Span {,, } Columnspace: Span {,,,, 5, 5 } b Find the nullspace and the columnspace of Nullspace: Span { 5,,,,, } Columnspace: Span {,, } c Let V denote the set of by matrices Show that trace: V R is linear What is the dimension of the kernel of trace? tracex + cy x ii + cy ii i x ii + c i y ii tracex + c tracey The dimension of the nullspace plus the dimension of the image must be nine, the dimension of the space of by matrices Since the image of trace is R, which has dimension one, the dimension of the kernel must be eight d Pick bases for V and R from c Find a matrix for trace using your bases Let A be the ordered basis i for V and let E be the standard basis for R The matrix M A,E is then the row matrix

7 REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS 7 x Some Notation for Solutions a Let V denote the set of by matrices trace: V R is linear Find a basis for the kernel of trace, and use it to find a general solution to tracev r + s + t for any real r, s or t b Let W denote the set of polynomials in x of degree or less Consider the linear function d d dx : W W Find a basis for the nullspace of dx and use it to find a general solution to d dx fx 4 x + rx + s for any real r or s xi Solving Problems in More Advanced Math Classes xii Inner Products a, b and c Which of the following x T y or x T 6 5 give inner products on R? Give evidence y or x T y All three satisfy the linearity in the first vector condition, because of the linearity of matrix multiplication The first, however, is not positive definite: The third is not symmetric: The second is symmetric because the matrix is but it still needs to be shown to be positive definite 6 x x y 6x 5 y + 4xy + 5y Consider the function hx 6x +4xy+5y where y is to be regarded as constant h x x+4y and h x So the horiontal tangent at x y is a minimum h y 6 y + 4 y y + 5y y This is obviously nonnegative, and can only be if y itself is But then hx only if x is too 6 x So x y > unless x, y is the ero vector So the second example 5 y is an inner product d Prove that f, g fxgxdx gives an inner product on the vector space of continuous functions on the unit interval

8 8 REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS Linearity and symmetry are basic properties of the integral and ordinary arithmetic applied to the integral: and fxgxdx fx + k hxgxdx gxfxdx gxfxdx + k hxgx dx It only remains to demonstrate the positivity condition If f is continuous and nonero at c [, ] there is some small interval of the form [a, b] near c so that for every point in this interval the magnitude of f is greater than a positive number ε So fx > ε > on this interval So by the definition of integral as a Riemann sum, fxfxdx b a fxfxdx > b a ε dx ε b a > We conclude that fxfxdx > unless f is the ero function e Prove that f, g fxgxdx does not give an inner product on the vector space of continuous functions on the interval [, ] A counterexample to the positivity condition is provided by the function that is between and, but has slope between and f Let V be the space of constant or first degree polynomials Define f, g on V to be fg + fg Is this an inner product? Yes: If ff+ff then both f and f are Since f is linear, that means f is the ero function The linearity and symmetry conditions follow quickly g Is v,w v v + v + w + w w an inner product on R? No: v, will be nonero if v, h Is v,w v w + v w + v w + v w an inner product on R? No: Positive definiteness fails though symmetry and the linearity condition do hold,,,

9 REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS 9 xiii The Matrix for an Inner Product Find the matrix G for the inner product on R given by x,y 7x y + x y + x y + x y G e,e e,e e,e e,e 7 xiv Orthogonal Complements Give R n the usual inner product in each case below a Find the orthogonal complement of Span {, 6,,, 4, 9,, } Solve the homogeneous system x y w The vectors,,, and 9, 4,, span this vector subspace, which is the orthogonal complement we are looking for b Find the orthogonal complement of Span {, 6,, 4, 9, } It is Span {,, } Use cross product c Find the orthogonal complement of Span {, 6 } It is Span { 6, } xv Orthonormal Basis a Let V be the space of constant or first degree polynomials Give V the inner product f, g fxgxdx Find an orthonormal basis What is the angle between x + and x? If you start{ with ordered basis {, x } and apply the orthonormaliation procedure you get, } x The angle between x + and x is about o b With V as above, define inner product f, g to be fg+fg Find an orthonormal basis What is the angle between x + and x? If you start{ with ordered basis {, x } and apply the orthonormaliation procedure you get, x } The angle between x + and x is about 7 o

10 REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS xvi Projection onto Subspaces in an Inner Product Space Let W Span {, 6,,, 4, 9,, } and give R 4 the usual inner product a Find the matrix with respect to the standard basis for the linear function Proj W hint: First find an orthonormal basis for W and extend it to an orthonormal basis for R 4 b Find the matrix, as above, for reflection Refl W of points in R 4 across W Apply the Gramm-Schmidt process to the ordered basis {, 6,,, 4, 9,,,,,,, 9, 4,, } and you get a particularly easy basis to calculate That is because the last two vectors span the orthogonal complement of the first two, cutting by two thirds the projections that must be calculated The basis you end up with consists of the ordered basis S {u,,u 4 } given by 47 6, 5499, 9 67, Proj W has a very simple matrix when expressed in these coordinates: M S,S This matrix kills the part of any vector that sticks out of the plane of W, but acts like the identity matrix in W However we want the matrix in terms of the standard basis E so we can work on standard coordinates Denoting by P S,E the matrix of transition from basis S to basis E we have P S,E [u ] E [u ] E [u ] E [u 4 ] E where the columns of this matrix are the standard coordinates of the new basis we calculated above The matrix we want is M E,E P S,E M S,S P E,S The matrix P E,S is the inverse of P S,E, but because both E and S are orthogonal, P S,E P T S,E The matrix K S,S for the reflection in W is given by K S,S This matrix preserves the part of a vector in W and reverses the part that sticks out of W It too can be transformed to K E,E, so it can work on standard coordinates, by the same process

11 REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS xvii Approximate Solutions a Find the point in R which is the best possible solution to 4 x y 7 6 where distance is measured with the usual inner product We first find the vector in the columnspace of the matrix closest to, 7, We apply the Gram-Schmidt procedure to the columns, yielding u,, 6 and u 6, 6, The projection of p, 7, onto the columnspace is p,u u + p,u u ,, 6 + 6, 6, , 95 8, 95 8 This is a point in the image of the linear transformation, so there is a solution, and it is the point in the image nearest to the target Now solve 4 x y 6 by the usual methods, yielding x and y 55 8 b Find the points in R which are best possible solutions to x 6 y 7 where distance is measured with the usual inner product The columnspace is spanned by, Projecting the target, 7 onto the columnspace yields the closest point v 5, 6 5 in the image of 6 Solving x 6 y yields x y t for any real t

12 REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS xviii Embedding an Inner Product Space in Euclidean Space Find an inner product on a finite dimensional vector space from the examples given above which are not dot product Find an orthonormal basis Satisfy yourself that the inner product corresponds to dot product on coordinates with respect to this basis We will examine the bases and inner products from xv Let V be the space of constant or first degree polynomials with ordered basis A {a, a } {, x } If we give V the inner product f, g { fxgxdx and apply Gramm-Schmidt we get basis C {c, c }, } x So the matrices of this inner product in these two bases are a,a G A a,a 45 and a,a a,a 45 9 c,c G C c,c c,c c,c If instead we endow V with inner product { f, g fg + fg we get an orthonormal basis D {d, d }, x } The matrices of this different inner product in these two bases are a,a H A a,a and a,a a,a d,d H D d,d d,d d,d xix Change of Basis Find matrices of transition from ordered basis A {, x } to orthonormal bases C for a and D for b from the last section Solutions: { a The matrix of transition from C P C,A so P A,C P { b The matrix of transition from D P D,A }, x to A {, x } is C,A, } x to A {, x } is so P A,D P D,A

13 REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS xx Effect of Change of Basis on the Matrix for an Inner Product Show that the matrix of transition can be used to turn the matrix of the inner product with respect to original basis into the identity matrix with respect to orthonormal bases in the two cases above The inner product becomes dot product on these coordinates P T A,D H D P A,D P T A,D I P A,D T H A xxi Effect of Change of Basis on the Matrix for a Linear Function Find the matrices M A,A, M C,C and M D,D corresponding to differentiation with respect to the three bases in xix Show how to use the matrices of transition to transform each matrix into the other two So d dx and M C,C P A,C M A,A P C,A d dx x so M A,A and M D,D P A,D M A,A P D,A xxii Basis of Eigenvectors a Let u, v and w be eigenvectors for three different eigenvalues for a linear function f Show that these three vectors form an independent set of vectors Suppose u, v and w are eigenvectors for distinct eigenvalues λ, λ and λ Relabel if necessary so that λ Suppose further that au + bv + cw for certain real numbers a, b and c Applying f to this sum and using linearity of f yields aλ u+bλ v +cλ w Dividing this by nonero λ and subtracting this linear combination from the earlier one yields b λ λ v + c λ λ w so b λ λ v c w λ λ

14 4 REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS Applying f to both sides of the rightmost equation yields different multiples of each side The only way that can happen is if both sides are the ero vector It follows that c and b are both, which implies that a too So the three eigenvectors are independent The same argument can be adapted to show that any finite number of eigenvectors for different eigenvalues must form an independent set of vectors hint: If this set is dependent there would be a dependency involving the fewest number of these vectors That number of vectors must exceed three, by our work from above So every coefficient in a linear combination of these particular eigenvectors which adds to the ero vector must be nonero Apply the argument from above to exhibit a nontrivial combination of these vectors involving fewer eigenvectors, yielding a contradiction and establishing the result b Find a basis of eigenvectors using the Cayley-Hamilton theorem as suggested in the notes for the matrix M M E,E where E is the standard basis of R c Find the matrix of transition from the standard basis E to the new basis B Verify that P E,B M E,E P B,E M B,B is diagonal The characteristic polynomial for this matrix is hx x x 5x 9 The columnspace of the matrices found below must be killed by the missing factor, and so are eigenvectors for the eigenvalue involved in the missing factor These calculations are so arduous to do by hand that hardware assistance is a practical necessity M IM 5I 6 64 M IM 9I 6 6 M 5IM 9I 4 We select ordered basis B of eigenvectors u,, and u,, and u 8, 8, These are eigenvectors for eigenvalues 9, 5 and, respectively

15 REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS 5 M B,B must be diagonal, and we verify M B,B P E,B M E,E P B,E d Suppose M A,A is a matrix for a linear transformation f : V V where V has dimension 4 and A {a,a,a,a 4 } We find that f has eigenvalues and 7 with corresponding eigenvectors v and v We also find that the vectors v, v, a, a in that order form a basis B What can you say about the matrix M B,B? The matrix will be [fv ] B [fv ] B [fa ] B [fa ] B The first two columns simplify nicely: the first is e and the second is 7e Nothing can be said about the last two columns