Q45E If v鈬赌聽聽is any nonzero vector... [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: Q45E (page 325)

If $\overset{鈬赌}{v}$ is any nonzero vector in $R^{2}$ , what is the dimension of the space Vof all $2 脳 2$ matrices for which $\overset{鈬赌}{v}$ is an eigenvector?

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.

Find the dimension

If $\overset{鈬赌}{v}$ is and non-zero vector in $R^{2}$ we want to find the dimension of the space V of all matrices for which $\overset{鈬赌}{v}$ is an Eigenvector.

Let A be a matrix in V .

Then let:

$A = [\begin{matrix} a & b \\ c & d \end{matrix}] a n d \overset{鈬赌}{v} = [\begin{matrix} v_{1} \\ v_{2} \end{matrix}]$

We will use the definition of eigenvector to see for with $\overset{鈬赌}{A v^{i}}$ an eigenvector.

$[\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} v_{1} \\ v_{2} \end{matrix}] = 位 [\begin{matrix} v_{1} \\ v_{2} \end{matrix}] [\begin{matrix} a v_{1} + b v_{2} \\ c v_{1} + d v^{2} \end{matrix}] = 位 [\begin{matrix} v_{1} \\ v_{2} \end{matrix}] \frac{v_{1}}{v_{2}} (a v_{1} + b v_{2}) = c v_{1} + d v_{2} \begin{matrix} \end{matrix}$

From here we can see that given any combination of 3 values $(a, b, c, o r d)$ , or, we could solve for the fourth unknown.

Thus, $d i m V = 3$ .

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91影视

If $\overset{鈬赌}{v}$ is any nonzero vector in $R^{2}$ , what is the dimension of the space Vof all $2 脳 2$ matrices for which $\overset{鈬赌}{v}$ is an eigenvector?

Short Answer

Step by step solution

Definition of the Eigenvectors

Find the dimension

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91影视

If v鈬赌is any nonzero vector in R2 , what is the dimension of the space Vof all 2脳2matrices for which v鈬赌is an eigenvector?

Short Answer

Step by step solution

Definition of the Eigenvectors

Find the dimension

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Calculus

Mechanics Maths

Theoretical and Mathematical Physics

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

If $\overset{鈬赌}{v}$ is any nonzero vector in $R^{2}$ , what is the dimension of the space Vof all $2 脳 2$ matrices for which $\overset{鈬赌}{v}$ is an eigenvector?