ANSWERS. E k E 2 E 1 A = B

Transcription

1 MATH 7- Final Exam Spring ANSWERS Essay Questions points Define an Elementary Matrix Display the fundamental matrix multiply equation which summarizes a sequence of swap, combination and multiply operations, transforming a matrix A into a matrix B An elementary matrix is a matrix E obtained from the identity matrix I by applying one combination, swap or multiply operation The equation is E k E E A = B where E, E,, E k are elementary matrices representing swap, multiply and combination operations that take A into B points State the Fundamental Theorem of Linear Algebra Include Part : The dimensions of the four subspaces, and Part : The orthogonality equations for the four subspaces Let A denote an m n matrix of rank r Part The dimensions of the nullspacea, colspacea, rowspacea, nullspacea T are respectively n r, r, r, m r Part rowspacea nullspacea, colspacea nullspacea T Both can be summarized by rowspace nullspace, applied to both A and A T Problems to 7 points Find a factorization A = LU into lower and upper triangular matrices for the matrix A =

2 Let E be the result of combo,,-/ on I, and E the result of combo,,-/ on I Then E E A = U = Let L = E E = 4 points Determine which values of k correspond to infinitely many solutions for the system A x = b given by 4 k A = k k 4, b = k There is a unique solution for deta, which implies k and k Elimination 4 k methods with swap, combo, multiply give k k Then Unique k k solution for three lead variables, equivalent to the determinant nonzero for the frame above, or k k ; No solution for k = [signal equation]; Infinitely many solutions for k = [one free variable] 5 points Find the complete solution x = x h + x p for the nonhomogeneous system x x x x 4 = The homogeneous solution x h is a linear combination of Strang s special solutions Symbol x p denotes a particular solution The augmented matrix has reduced row echelon form last frame equal to the matrix Then x = t, x =, x =, x 4 = t is the general solution in scalar

3 form The partial derivative on t gives the homogeneous solution basis vector The partial derivative on t gives the homogeneous solution basis vector Then x h = c + c Set t = t = in the scalar solution to find a particular solution x p = 6 points Define S to be the set of all vectors x in R such that x + x = and x + x = x Apply a theorem which concludes that S is a subspace of R This implies stating the hypotheses and checking that they apply Let A = Then the restriction equations can be written as A x = Apply the nullspace theorem also called the kernel theorem, which says that the nullspace of a matrix is a subspace Another solution: The given restriction equations are linear homogeneous algebraic equations Therefore, S is the nullspace of some matrix B, hence a subspace of R This solution uses the fact that linear homogeneous algebraic equations can be written as a matrix equation B x =, without actually finding the matrix 7 points The 5 7 matrix A below has some independent columns Report the

4 independent columns of A, according to the Pivot Theorem A = 6 6 / Compute rrefa = The pivot columns are and points Let S be the subspace of R 4 spanned by the vectors v = v = Find the Gram-Schmidt orthonormal basis of S and Let y = v and u = y y Then u = Let y = v minus the shadow projection of v onto the span of v Then y = v v v v = v v 4

5 Finally, u = y y We report the Gram-Schmidt basis: u =, u = 5 9 points Define matrix A and vector b by the equations A = 4, b = Find the value of x by Cramer s Rule in the system A x = b x = /, = det 4 = 6, = deta = 4, x = 9 points Display the entry in row 4, column of the adjugate matrix of A = 4 The adjugate matrix is also called the adjoint matrix It appears in the formula C = adjc detc The answer is the cofactor of A in row, column 4 = 7 times the minor of A in row, column 4 The minor equals 8 The answer is 8 points Consider a real matrix A with eigenpairs 5, 6,,,, 4 Display an invertible matrix P and a diagonal matrix D such that AP = P D 5

6 The columns of P are the eigenvectors and the diagonal entries of D are the eigenvalues, taken in the same order points Find the eigenvalues of the matrix A = 5 To save time, do not find eigenvectors! The characteristic polynomial is deta ri = r rr r + The eigenvalues are,,, Determinant expansion of deta λi is by the cofactor method along column This reduces it to a determinant, which can be expanded by the cofactor method along column points Let I denote the identity matrix Assume given two matrices B, C, which satisfy CP = P B for some invertible matrix P Let C have eigenvalues,, Let A = I + B Find the eigenvalues of A Both B and C have the same eigenvalues, because detb λi = detp B λip = detp CP λp P = detc λi Further, both B and C are diagonalizable The answer is the same for all such matrices, so the computation can be done for a diagonal matrix B = diag,, In this case, A = I + B = diag,, + diag,, = diag, 5, and the eigenvalues of A are, 5, To finish, observe that the eigenvalues of A and A are related, because A v = λ v implies A v = λ v Therefore, the eigenvalues of A are, 5, 8 4 points The Cayley-Hamilton theorem suggests that there are real matrices A such that A = A Give an example of one such matrix A Choose any matrix whose characteristic equation is λ λ = Then A A = by the Cayley-Hamilton theorem 5 4 points The spectral theorem says that a symmetric matrix A can be factored into A = QDQ T where Q is orthogonal and D is diagonal Find Q and D for the symmetric 6

7 matrix A = Start with the equation r 4r + = having roots r =, Compute the eigenpairs, v,, v where v = and v = The two vectors are orthogonal but not of unit length Unitize them to get u = v, u = v Then Q =< u u >=, D = diag, 6 points Let the linear transformation T from R to R be defined by its action on two independent vectors: T 4 4 =, T = 4 Find the unique matrix A such that T is defined by the matrix multiply equation T x = A x Let A be the matrix for T, such that T v = A v matrix multiply Then A Solving for A gives A = = 4/ 4/ 4/ / = 7 points Assume singular value decomposition A = UΣV T given by = 8 T Find a formula for the pseudo-inverse, but don t bother to multiply out matrices The definition is A + = V Σ + U T Identifying the matrices from the formula gives A + = T 8 7