Math 308 Final Exam Winter 2015, Form Bonus. of (10) 135

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1 Math 308 Final Exam Winter 015, Your Name Your Signature Student ID # Points Form Bonus of (10) 135 No books are allowed. But you are allowed one sheet (10 x 8) of handwritten notes (back and front). You may use a calculator. Place a box around your final answer to each question. If you need more room, use the back of each page and indicate to the grader how to find the logic order of your answer. Raise your hand if you have questions or need more paper. For TRUE/FALSE problems, you just need to cross the right box. For each correct answer, you will get 1 point, for each incorrect answer, -1 point is added. For no answer you will get zero points. In each subsection of the TRUE/FALSE part, you can never get less than zero points. In order to get points for formal correctness, underline vectors, use {}-brackets for sets, declare parameters, mark equivalent matrices properly and keep a reasonable order and neatness. Do not open the test until everyone has a copy and the start of the test is announced. GOOD LUCK!

2 1.) For each correct answer in the TRUE/FALSE part, you will get 1 point, for each incorrect answer, there will be one point subtracted, i.e. you get -1 point. For no answer, you get 0 points. You can not get less than 0 points out of one subproblem (which are the problems, (a)-(h)) (a) (b) (c) Cross the right box for the statements about linear systems. A homogeneous system can either have no solution, a unique solution TRUE or infinitely many solutions. It is possible that an inhomogeneous system does not have a solution. TRUE A homogeneous system with 5 variables and 5 equations has exactly TRUE one solution. If s 0 is a solution to a linear system of the form Ax = 0 for a TRUE matrix A, then this system has infinitely many solutions. A homogeneous system with at least one free variable has infinitely TRUE many solutions. Let A be a (5, 7)-matrix. Then any solution to Ax = 0 is a vector in TRUE R 7. Let A be a (3, 3)-matrix with linearly independent rows. Then Ax = b has exactly one solution for any b in R 3. TRUE Cross the right box for the statements about span. Let S := {u 1, u, u 3, u 4 } R n. Then u 1 u 1 + 5u 0u 4 is an TRUE element of span(s). The span of {u} has infinitely many elements for any choice of u R n. TRUE {u 1, u, u 3, u 4 } R 3 spans R 3 for any choice of vectors u i. TRUE Let {u 1, u,..., u m } R n. Then span{u 1, u,..., u m } {u 1, u 1 + TRUE u, u, u 3,..., u m }. Let A be an (n, m)-matrix and let u col(a). Then Ax = u has a TRUE solution. Let 0 u be a vector in R n. Then 0 span{u}. TRUE Let u 0 span{u 1,..., u m } R n. Then {u 0, u 1, u,..., u m } is linearly TRUE dependent. Let A be an (m, n)-matrix. Then {Au u R n } = col(a). TRUE Cross the right box for the statements about linear independence, span, bases and dimensions. For any vector u and a R, the set {u, au} is linearly dependent. TRUE Let S = span{u 1, u }. Then dim(s) =. TRUE Let S R 4 be a subspace of dimension 3. Then S has a uniquely TRUE determined basis with 4 elements. Let S 1 S be subspaces of R n. Then dim(s 1 ) < dim(s ). TRUE Let S be a subspace of R n with dim(s) = m and let U := TRUE {u 1,..., u k } S. If k > m then U is linearly dependent. {u 1, u, u 3 } R is a linearly dependent set. TRUE Let A be an (n, m)-matrix. Then null(a) is a subspace of R m. TRUE Let A be a matrix. If nullity(a) = 3, then Ax = 0 has infinitely TRUE many solutions.

3 (d) Cross the right box for the statements about matrices and homomorphisms. Let T be a homomorphism with corresponding matrix A T. If TRUE nullity(a T ) 1 then T is not injective. Let T : R 4 R 8 be a homomorphism. Then T can be surjective, TRUE but not injective. Let T be a homomorphism with corresponding matrix A T. TRUE Then ker(t ) = null(a T ). Let T : R 1 R 4. Then dim(ker(t )) must be 8 or greater. TRUE Let T : R m R n be a homomorphism. Then range(t ) R n. TRUE The function T : R m R m, u u is a linear homomorphism. TRUE If T : R m R n is a surjective homomorphism with corresponding TRUE matrix A T, then b col(a T ) for any b R n. If T is an isomorphism with domain R n and corresponding matrix A T, then A T is an (n, n)-matrix. TRUE (e) Cross the right box for the statements about matrices. Let A, B be (n, n)-matrices. Then A B = (A+B)(A B) TRUE Let A be an (n, m)-matrix. Then A is defined. TRUE Let A be a square matrix. Then det(a) = det(a). TRUE A square matrix A is singular, if and only det(a) = 0. TRUE If A is invertible and Ax = b has a solution s, then b is a TRUE solution to A 1 x = s. Let A, B be equivalent matrices. Then det(a) = det(b). TRUE (f) Cross the right box for the statements about column- and rowspace of a matrix A. Let A, B be equivalent matrices. Then row(a) = row(b). TRUE The rank of a matrix A is equal to the dimension of the row TRUE space of A. Let A be a matrix. Then row(a) = col(a). TRUE Let A be an (m, n)-matrix. Then rank(a) is less than or equal to m. TRUE (g) Cross the right box for the statements about eigenvalues and eigenspaces of an (n, n)-matrix A. Any vector 0 u R n that satisfies Au = λu for some λ R TRUE is an eigenvector of A. Let λ be an eigenvalue for A. Then the set of eigenvectors of TRUE A with eigenvalue λ forms the eigenspace E λ of A. The zero vector is always an eigenvector for any eigenvalue TRUE for A because it satisfies the defining property A0 = λ0. If rank(a) is less than the number of columns of A, then 0 is TRUE an eigenvalue of A. The matrix A may not have an eigenvalue. TRUE (h) Cross the right box for the statements about orthogonality of vectors. Let u 1, u be orthogonal to u, then u 1 + u is also orthogonal TRUE to u. Two vectors in R 1 can only be orthogonal if at least one of TRUE them is the zero vector. There is no nonzero vector u that is orthogonal to u. TRUE Let A be a matrix and u null(a). Then u row(a). TRUE

4 .( points) Consider the following linear homomorphism: u 1 u 1 + 3u + u 3 T : u u 3 u + u 3 u 1 3u 3. (a) Find the corresponding matrix A, such that T (x) = Ax for all x R 3. (b) Find the kernel of T. (c) Calculate the determinant of A. Is T invertible? Justify your answer. (d) Find the inverse T 1 of T using A.

5 3.(4++4+ points) Consider the following matrix: 1 3 A = (a) Find the null space of A. (b) Verify that the vector v = [0, 1, 0,, 1] t is an element of null(a). (c) Find a basis of null(a), that contains the vector v = [0, 1, 0,, 1] t. (d) What is the nullity of A? What is the rank of A?

6 4.( points) Consider the following matrix: (a) Determine the characteristic polynomial χ A of A. Show all your work! (Key, so that you can continue: The characteristic polynomial is χ A = λ 4 4λ 3 + 5λ λ.) (b) Determine the eigenvalues for A. (c) Compute the eigenspace for the eigenvalue λ = 1. eigenspace? What is the dimension of this (e) What is the only possible dimension of the eigenspace with eigenvalue λ = 0? Answer this question with the help of χ A and justify your answer. (f) Based on the knowledge about the eigenvalues of this matrix, what can be said about the determinant of A?

7 5.(4++ points) Let S = span{s 1 = 1 4 0, s = 0 4, s 3 = }. (a) Find a basis for S. (b) Compute the norm s 1 of s 1. (c) What is the norm of 1 s 1 s 1? 6. (3 points) Find a matrix that has χ = λ + as its characteristic polynomial.

8 7.( points) Suppose that A is a (5, 16)-matrix. (a) What is the maximum possible value for rank(a)? (b) What is the minimum possible value for nullity(a)? (c) Suppose that dim(col(a)) = 5. What is nullity(a)? (d) Consider the homomorphism T : R 16 R 5, x Ax. (i) What does the nullity of A represent in terms of T? (ii) What is the dimension of the range of T if the nullity of A is at its minimum value? Is T then surjective? 8.(4+1+1 points) Let S R 5 be a subspace of dimension 4. (a) What are the possible dimensions of subspaces S i, that are subsets of S, i.e. S i S? (b) How many elements does the subspace of S of dimension 0 have? (c) How many elements does a subspace of S of dimension 1 have? 9.(3 points) Calculate ([ ] + [ 1 1 ]).

9 [ 3 10.(4 points) Let B 1 = { 1 ] [ 5, 1 ] [ 8 } be a basis for R and let u = ] be a vector represented with respect to the standard matrix. What is the coordinate vector of x with respect to B 1? 11.(3+3 points) Let A = (a) Have a close look at A and find its determinant without actually computing it.. (b) Is λ = 0 an eigenvalue of A? Justify your answer.

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