 # Linear Algebra Problems

## Transcribed Text

1. Let V = M2×2(R), and let T  a b c d =  d b c a be a linear operator on V . Find the eigenvalues of T, and an ordered basis β for V such that [T]β is a diagonal matrix. 2. Prove that the eigenvalues of an upper triangular matrix A are the diagonal entries. 3. Prove that similar matrices have the same characteristic polynomial. 4. Two matrices A, B ∈ M2×2(R) are simultaneously diagonalizable if there exists an invertible matrix Q ∈ M2×2 such that both Q−1AQ and Q−1BQ are diagonal matrices. Prove that if A and B are simultaneously digonalizable matrices, then A and B commute (i.e. AB = BA). 5. Let T be a linear operator on an inner product space V , and suppose that ||T(x)|| = ||x|| for all x. Prove that T is one-to-one. 6. Consider the inner product space V = R 3 . Let S = (1, 0, 1),(0, 1, 1),(1, 3, 3). (a) Use the Gram-Schmidt process on S to obtain an orthogonal basis β for Span(S). (b) Normalize the vectors in β to obtain an orthonormal basis β 0 for S. (c) Compute the Fourier coecients of x = (1, 1, 2) relative to β

## Solution 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. \$20.00 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.

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