1. Select the unique answer (choice) for each problem. Write only the answer.

Transcription

1 MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x + x x = x + 7x + 5x = 9 x + x a x = a A. a = only B. a = only C. a = only D. a = or a = E. a = and a = C a a = = 8 a + a 8 a + 9 a () Let A be a 4 real matrix. Which of the following statements must be true? (i) rank(a) = (ii) The columns of A are linearly dependent (iii) The null space of A has infinitely many nonzero vectors (iv) Ax = b has a unique solution for any b R A. (i) and (ii) only B. (ii) and (iii) only C. (i), (ii), and (iii) only D. (ii), (iii), and (iv) only E. All of them B () Which of the following subsets S is a subspace of V? A. V = M(, ) and S is the set of all real matrices with determinants equal to B. V = M(n, n) and S is the set of all n n diagonalizable matrices The problems do not necessarily or accurately reflect the contents, scope, or depth of the exams. Page of 9

2 C. V = R and S is the set of vectors [x, y, z T satisfying x + y z = D. V = R and S is the set of vectors [x, y T satisfying x + y = E. V is the set of all real polynomials of degrees at most (including the zero polynomial), and S is the set of all real polynomials of the form ax + D (4) Which of the following sets of vectors forms a basis for R? A., B.,, C.,, D.,, E.,,, D. It needs exactly linearly independent vectors. (5) If A is an n n matrix. Which of the following statements is true? A. If B is constructed from A by interchanging the first two rows of A, then det B = det A B. If det A =, then A has a zero row or a zero column C. If det A =, then some two rows of A are proportional D. If some two rows of A are proportional, then det A = E. None of the above D (6) Let A = [ which of the following sets gives a basis of the null space of A? A. B., Page of 9.

3 C. D.,,, E. None of the above B. There should be two vectors in the null space since the rank of A is. (7) Suppose A is a matrix with an eigenvector x corresponding to an eigenvalue equal to. Which of the following statements is wrong? A. A must be singular B. Ax = C. A x = D. A must be deficient E. The columns of A must be linearly dependent D (8) Which of the following matrices are diagonalizable? A = [ A. A only B. A and B only C. C and D only D. A, B, and C only E. All of them B [, B = [, C = [, D =. Let V be the set of all real multiples of exponential functions of the form e kx, where k R. Decide if V is a vector space under the following element addition and scalar multiplication: respectively. c e kx c e mx = c c e (k+m)x, r c e kx = rc e kx, Suppose for any v = c e kx V, c e kx = c e kx for = c e mx. Then we must have c = and m =. That is, =. However, for V, there does not exist v. Thus, V is not a vector space... Consider A = Page of 9

4 () Find a basis for the row space of A. () Find a basis for the column space of A. () Find a reduced row echelon form for A. (4) What is the nullity of A? What is the dimension of W, where W is the row space of A? (5) Find a basis for W = = A basis for W is { [ 4 5, [ 6 6, [ }. 4 A basis for W is,,. To get a RREF: = 6 = The nullity of A is 5 =. W consists of all vectors x satisfying Ax =. Thus, dim W is the nullity of A. x + x 5 x x = x 5 = x + x 5 A basis for W is x 5, 4. Let W be the column space of A = () Find an orthogonal basis for W.. Page 4 of 9

5 () Let x =. Compute proj W x and orth W x. () Find a QR factorization of A. Let A, A, A be the three columns of A. p = A =, p = A A p p = p p p = A A p p p p A p p p p = 5. Consider A = A = = = () Suppose Ax = By Cramer s rule, p p p = 5 5. Find x (the second entry of x). x = / () Show that A is invertible and find A. det A = 8 Page 5 of 9. = 8 = 4 =

6 () Find det(a ). det(a ) = det A = (4) Find all the eigenvalues and eigenvectors of A. 4 λ =, x =, λ =, y = 7 5 (5) Determine if A is diagonalizable. If yes, diagonalize it. A = P DP, λ = 4, z =. where P = (6) Compute A b, where b = b = x + y + z, so , D = 4 A b = A (x + z) = λ x + λ z = + 4 = (7) Find all the eigenvalues and eigenvectors of A. For Ax = λx, since A exists, we get A x = λ x Thus, the eigenvalues are those of the inverses of the eigenvalues of A, with the same eigenvectors. 6. Find the eigenvalues and eigenvectors of A = [, and for each eigenvalue λ, Page 6 of 9

7 find λ. λ = + i, x = λ = i, y = [ i [ + i, λ = (cos π + i sin π ) = cos π π + i sin = i λ = λ = + i 7. Let A be an m n real matrix and let W R n be the nullspace of A. Prove that for all x R n, Ax = A(orth W x). orth W x = x proj W x A(orth W x) = Ax A proj W x Since proj W x is in the null space of A, A(proj W x) =. 8. Let θ be the angle between x = and y =. () Find cos θ. () Let W = span{x, y} and z = 4. Find orth W z. cos θ = 4 x, y are not orthogonal. To find orth W z, we orthogonalize x, y first. 9. Let C([, ) be the vector space of continuous functions on the interval [,, and define the transformation T : C([, ) R by T (f) = Show that T is a linear transformation. Page 7 of 9 f(x)dx.

8 For a, b R and f(x), g(x) C([, ), T (af(x) + bg(x)) = (af(x) + bg(x))dx = at (f) + bt (g). Let P be the vector space of real polynomials of degrees at most (including the zero polynomial). Consider a linear transformation T : P P satisfying Find T ( + x + x ). T ( + x) = x, T (x x ) = x, T ( + x + x ) = + x. + x + x = ( + x) (x x ) + ( + x + x ) Thus, T ( + x + x ) = x ( x) + + x = x + x. Determine if the following statements are true, and justify your answers. () If A is an n n matrix with two equal rows, then there exists a nonzero vector x such that A x =. T. The columns of A are linearly dependent. () If matrices A, B, C satisfy AB = AC and A, then B = C. F () If A, B are n n matrices and C = A + B, then det(a) + det(b) = det(c). F (4) A set of orthonormal vectors in R n is linearly independent. T. Since a set of orthonormal vectors is also orthogonal, and a set of orthogonal vectors is linearly independent. (5) The following matrices are linearly independent: [ [ [ A =, B =, C = [, D =. T det = (6) Let W be a subspace of R n. Then dim W + dim W = n. T. Suppose {p,..., p k } is a basis for W. Let A be a matrix with the rows given by all p T i. Then W is the row space of A, and W is the null space of A. Then the result follows from the rank-nullity theorem. Page 8 of 9

9 (7) A subset of a linearly independent set is linearly independent. T ([ ) [ a a + b (8) Let T : R R defined by T = b ab. Then T is a linear transformation. F Page 9 of 9