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Linear Algebra With Applications
Found in: Page 324
Linear Algebra With Applications

Linear Algebra With Applications

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 2×2 matrices A for which both [11] and [12] are eigenvectors, and thus determine the dimension of .

Hence, the required dimension is 2.

See the step by step solution

Step by Step Solution

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 2×2 matrix whose columns are the vectors 11 and 12.


Then, the matrix A can be calculated as,


Here, D is the diagonal matrix.

Let it be: D=a00b, where, a and b are the eigenvalues of A.

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

S-1=1ad-bcd-b-ca =12-12-1-11 =2-1-11

Step 3: Finding dimension

Now, the matrix A can be written as,


Thus, a basis of the linear space V is 2-12-1,-11-22, and from this it is clear that the dimension of V is dim V = 2.

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