Q39E Find a basis of the linear space... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 7: Q39E (page 324)

Find a basis of the linear space Vof all $2 脳 2$ matrices Afor whichrole="math" localid="1659530325801" $[\begin{matrix} 0 \\ 1 \end{matrix}]$ is an eigenvector, and thus determine the dimension of V.

Short Answer

Expert verified

Hence, the required dimension is 3.

Step by step solution

Definition of the Eigenvectors

Eigenvectors are a nonzero vector that is mapped by a given linear transformation of a vector space onto a vector that is the product of a scalar multiplied by the original vector.

Finding eigenvalues

If $[\begin{matrix} 0 \\ 1 \end{matrix}]$ is an Eigen vector of A then we know that $A \overset{鈫�}{x} = 位 \overset{鈫�}{x}$ , thus we can write:

$[\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} 0 \\ 1 \end{matrix}] = 位 [\begin{matrix} 0 \\ 1 \end{matrix}]$

and we must solve for A.

We will have:

$[\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} 0 \\ 1 \end{matrix}] = 位 [\begin{matrix} 0 \\ 1 \end{matrix}] [\begin{matrix} b \\ d \end{matrix}] = 位 [\begin{matrix} 0 \\ 1 \end{matrix}]$

Thus, A can be any matrix of the form: $[\begin{matrix} a & 0 \\ c & d \end{matrix}]$ . From here we can see that a basis for A is all lower triangular matrices and thus the dimension is 3.

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91影视

Find a basis of the linear space Vof all $2 脳 2$ matrices Afor whichrole="math" localid="1659530325801" $[\begin{matrix} 0 \\ 1 \end{matrix}]$ is an eigenvector, and thus determine the dimension of V.

Short Answer

Step by step solution

Definition of the Eigenvectors

Finding eigenvalues

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91影视

Find a basis of the linear space Vof all 2脳2matrices Afor whichrole="math" localid="1659530325801" [01]is an eigenvector, and thus determine the dimension of V.

Short Answer

Step by step solution

Definition of the Eigenvectors

Finding eigenvalues

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Theoretical and Mathematical Physics

Probability and Statistics

Discrete Mathematics

Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Find a basis of the linear space Vof all $2 脳 2$ matrices Afor whichrole="math" localid="1659530325801" $[\begin{matrix} 0 \\ 1 \end{matrix}]$ is an eigenvector, and thus determine the dimension of V.