Transcribed TextTranscribed Text

I. (10pts) Find all the solutions of the following system of equations (w/o a calculator). T1 + T2 + 3.3 = 0 2.11 + T2 + 4.33 = 1 3.11 + T2 + 5.3 = 2 II. (10pts) T + x2 (a) Show that B is a basis for the space P2 of polynomials of degree less or equal to 2. (b) Write p(x) = 1 + 3.c + 2² as a linear combination of vectors in B. III. (10pts) Let I : R² R² be the identity operator (i.e. I(v) = v for any U € R². Let 1 1 S = { } and T = { -1 D -1 , , }. Find the matrix representation of 1 2 I with respect to (a) S; (b) T; (c) S and T; (d) T and S. IV. (10pts) Let A be the matrix Is A diagonalizable? If yes, find the 0 0 2 0 1 0 0 2 diagonal matrix similar to A. V. (20pts) Let P2 denote the space of polynomials of degree less then or equal to 2. Let T : P2 P2 be defined as T(f) = 2f' + f". (a) Show that T is a linear transformation. (b) Let S = {1,t,t2} Find the matrix representation A of T with respect to the ordered basis S. (c) Find a basis for the kernel of T. and a basis for the image of T. (d) Write the characteristic polynomial of the matrix A from part (b). Find the eigenvalues of A. (e) For each eigenvalue in part (d) find a basis for the corresponding eigenspace. (f) Is A diagonalizable? VI. (16pts) For each statement below answer whether it is TRUE or FALSE. 1. If a vector V is orthogonal to vectors u and W, then it is orthogonal to each vector in the span (u, w). 2. Any set of vectors containing the zero vector is linearly dependent. 3. det = det B. 4. If a linear transformation L : R³ R³ is onto, then it is invertible. 5. If A is a 3 X 5 matrix whose null space has dimension 2, then the equation Ax ==== 0 has infinitely many solutions. 6. Every subspace of Rn has at least one orthonormal basis. 7. If A is a 3 X 3 matrix with eigenvalues / then it is diagonalizable. 8. The determinant of an elementary matrix is always 1.

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

    or FREE if you
    register a new account!

    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