Q41E Find a basis of the linear spac... [FREE SOLUTION]

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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 7: Q41E (page 324)

Find a basis of the linear space V of all $2 脳 2$ matrices Afor which both $[\begin{matrix} 1 \\ 1 \end{matrix}]$ and $[\begin{matrix} 1 \\ 2 \end{matrix}]$ are eigenvectors, and thus determine the dimension of.

Short Answer

Expert verified

Hence, the required dimension is 2.

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 inverse of matrix

Let, Sbe a $2 脳 2$ matrix whose columns are the vectors $[\begin{matrix} 1 \\ 1 \end{matrix}]$ and $[\begin{matrix} 1 \\ 2 \end{matrix}]$ .

$S = [\begin{matrix} 1 & 1 \\ 1 & 2 \end{matrix}]$

Then, the matrix A can be calculated as,

S'AS=D

Here, D is the diagonal matrix.

Let it be: $D = [\begin{matrix} a & 0 \\ 0 & b \end{matrix}]$ , where, aand b are the eigenvalues of A.

First, compute the inverse of the matrix S as follows:

$S^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}] = \frac{1}{2 - 1} [\begin{matrix} 2 & - 1 \\ - 1 & 1 \end{matrix}] = [\begin{matrix} 2 & - 1 \\ - 1 & 1 \end{matrix}]$

Finding dimension

Now, the matrix A can be written as,

$A = S D S^{1} = [\begin{matrix} 1 & 1 \\ 1 & 2 \end{matrix}] [\begin{matrix} a & 0 \\ 0 & b \end{matrix}] [\begin{matrix} 2 & - 1 \\ - 1 & 1 \end{matrix}] = [\begin{matrix} 1 & 1 \\ 1 & 2 \end{matrix}] [\begin{matrix} 2 a & - a \\ - b & b \end{matrix}] = [\begin{matrix} 2 a - b & - a + b \\ 2 a - 2 b & a + 2 b \end{matrix}] = [\begin{matrix} 2 a & - a \\ 2 a & - a \end{matrix}] + [\begin{matrix} - b & b \\ - 2 b & 2 b \end{matrix}] = [\begin{matrix} 2 & - 1 \\ 2 & - 1 \end{matrix}] a + = [\begin{matrix} - 1 & 1 \\ - 2 & 2 \end{matrix}] b$

Thus, a basis of the linear space Vis $\{[\begin{matrix} 2 & - 1 \\ 2 & - 1 \end{matrix}], [\begin{matrix} - 1 & 1 \\ - 2 & 2 \end{matrix}]\}$ , and from this it is clear that the dimension of Vis dim V = 2.

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91影视

Find a basis of the linear space V of all $2 脳 2$ matrices Afor which both $[\begin{matrix} 1 \\ 1 \end{matrix}]$ and $[\begin{matrix} 1 \\ 2 \end{matrix}]$ are eigenvectors, and thus determine the dimension of.

Short Answer

Step by step solution

Definition of the Eigenvectors

Find inverse of matrix

Finding dimension

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91影视

Find a basis of the linear space V of all 2脳2matrices Afor which both[11]and[12]are eigenvectors, and thus determine the dimension of.

Short Answer

Step by step solution

Definition of the Eigenvectors

Find inverse of matrix

Finding dimension

One App. One Place for Learning.

Most popular questions from this chapter

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Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Find a basis of the linear space V of all $2 脳 2$ matrices Afor which both $[\begin{matrix} 1 \\ 1 \end{matrix}]$ and $[\begin{matrix} 1 \\ 2 \end{matrix}]$ are eigenvectors, and thus determine the dimension of.