Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

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

Hence, the required dimension is 2.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: 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.

Step 2: Find inverse of matrix

Let, S be a $2 \times 2$ 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, a and b are the eigenvalues of A.

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

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

Step 3: 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 V is $\{[\begin{matrix} 2 & - 1 \\ 2 & - 1 \end{matrix}], [\begin{matrix} - 1 & 1 \\ - 2 & 2 \end{matrix}]\}$ , and from this it is clear that the dimension of V is dim V = 2.

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Linear Algebra With Applications

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Short Answer

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

Step by Step Solution

Step 1: Definition of the Eigenvectors

Step 2: Find inverse of matrix

Step 3: Finding dimension

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Linear Algebra With Applications

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Step by Step Solution

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