QuestionQuestion

Transcribed TextTranscribed Text

1. (a) Define that T U - V is a linear map from the vector space U to the vector space V. (b) Assume that are dependent in U. Prove that T(r)) T(x) are linearly dependent in V. 2. (a) Let T:U - V be linear and W be a subspace of V. Prove that '(W) = X = {=|T(a) E w} is a subspasce of U. (b) Prove that dirn(X) dim(W). 3. State the definition of the null space and of the range of a linear map T. 4. Let U be a vector space of dimension 2n, that is, U has a basis v1, 02, Wn- Let T be the linear map T:U - U such that - - and 0,1=1, n. What can you say about dim(null-space(T)) and din(range(T))? Prove that null-space(T) = range(T). 5. Prove that the solution space of a linear homogemeous system AX = 0 of m equations in n unknowns with real coefficients is a subspace of R". 6. Let T: R - R™ be linear where m < n. Prove that the null-space of T is nom-trivial, that is different from {0}. 7. Prove that a linear map T:U - V is injective if and only if the null-space is trivial. 8. Let T be as linear map from R= to Rm. How is the matrix A for T with respect to the unit vectors defined? Write down the matrix for the linear map T in problem 4. 9. Express mul-space(T) and range(T) in terms of its matrix A. Relate the dimensions of these spaces to row-rank and column-rank of A. 10. Let T be a linent map from R= - R and let A € R (a) Define that A is an eigenvalue of T. (b) Define that E2 is the eigenspace for &

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

Linear Algebra Problems
    $40.00 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.

    Decision:
    Upload a file
    Continue without uploading

    SUBMIT YOUR HOMEWORK
    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