Transcribed TextTranscribed Text

1. The augmented matrix of a linear system has the form 1 2 1 a -1 3 1 1 b 3 -5 5 1 C 2 -2 4 2 d (a) Can you determine by inspection whether the determinant of the coefficient matrix is 0? Explain. (b) Can you decide by inspection whether the linear system has a unique solu- tion for every choice of a, b, C. d? Explain. (c) Determine the values of a, b, c. d for which the linear system is consistent and inconsistent. (d) If a - 2, b = 1. C - -1, and d - 4, find the solution set for the linear system. 2. Let 1 1 M1 = 1 M2 = 2 1 M3 = 1 : (a) Show that the set {M1. M2, Ms} is linearly independent. (b) Describe all matrices that can be written as a linear combination of M1, M2, and M3. (c) Find a basis for the space of matrices found in part (b) and give the space's dimension. (d) Give an example of a space that is isomorphic to the space found in (b). Specify the isomorphism between the two spaces. 3. Let c be a fixed scalar and let m(x) 1 p((c)==to (3(1)=(I+C)? = (a) Show that B - {p1(x) p2(x), P3(x)} is a basis for P2. (b) Find the coordinates of f(x) = ao + ax + 02x² relative to B. 4. Let W C R3 (i.e. W is a subset of R³) be the plane with equation I- = 21 i.e. W = {F € R3 on - T2 - T3 - 0} where Fil = T2 T3 (a) Show W is a subspace of the vector space R³ with standard addition and scalar multiplication (b) Find a basis for W and then use the Gram-Schmidt procedure to obtain an orthonormal basis for W. 5. Define a transformation T P2 R by T(p(x)) = fo p(c)de (a) Show T is a linear transformation (b) Compute N(T). Is T one-to-one? (c) Show that T is onto. (d) Let B be the standard basis for P2 and let B' = {1} be a basis for R. Find [T), (e) Use the matrix found in part (d) to compute T1-x2 - - 3x+2). a 6. Let A = for some a. 6 € R. b a (a) Find the eigenvalues and eigenvectors of A. (b) Diagonalize A. That is, specify P and D such that D = P-1AP. 7. Let A be an n x n matrix. Show that det(A7) = det(A) using a proof by induction. (Hint: the base case is not n=1.) 8. Suppose that S = {01,02,23} is linearly independent and W1 = is + in + is 12- w3 U3. Show that T - {ur W2 w3} is linearly independent.

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:

    for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Linear Algebra Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats