5. Which of the following are true and which false? Give reasons for each of your answers. (a) It is possible to find a pair of five-dimensional subspaces S and T of R® such that SNT = {0}. (b) If none of the eigenvalues of a square matrix is 0, then it is invertible. (c) If an n by n matrix does not have n distinct eigenvalues then it is not diagonalizable.

5. Which of the following are true and which false? Give reasons for each of your answers. (a) It is possible to find a pair of five-dimensional subspaces S and T of R® such that SNT = {0}. (b) If none of the eigenvalues of a square matrix is 0, then it is invertible. (c) If an n by n matrix does not have n distinct eigenvalues then it is not diagonalizable.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

**Linear Algebra Problem Set**

**Problem 5: True or False Analysis**

Analyze the following statements and determine which are true and which are false. Provide justifications for each of your answers.

(a) **It is possible to find a pair of five-dimensional subspaces $ S $ and $ T $ of $\mathbb{R}^8$ such that $ S \cap T = \{0\} $.**

(b) **If none of the eigenvalues of a square matrix is 0, then it is invertible.**

(c) **If an $ n $ by $ n $ matrix does not have $ n $ distinct eigenvalues, then it is not diagonalizable.**

(d) **Let $ M $ be a 2 by 2 invertible matrix. Let $ S $ be the set of all 2 by 2 matrices $ A $ that are diagonalized by $ M $. That is, $ M^{-1}AM $ is diagonal for any such matrix $ A $ in the set $ S $. True or False: $ S $ is a subspace of the vector space of 2 by 2 matrices.**

(e) **Let $ M $ be a 2 by 2 invertible matrix which diagonalizes both $ A $ and $ B $. Then $ M $ does not necessarily diagonalize $ AB $.**

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