hw6sol - Winter 2011 Math 67 Linear Algebra Homework 6...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Winter 2011 Math 67 Linear Algebra Homework 6 Problems 5.9, 5.14 and 5.19 from Axler. Solution. (5.9) Suppose that T : V V is a linear map with rank k . If v is an eigenvector with a non-zero eigenvector then T ( v/ ) = v . This implies that v is in the range of T . There are at most k distinct non-zero eigenvalues of T , since each such eigenvector is in the range (which is k-dimensional), and eigenvectors with distinct eigenvalues are linearly independent. Taking into account that 0 is a possible eigenvalues, one sees that there are at most k + 1 distinct eigenvalues of T . Solution. (5.14) We only need to check that this for p ( z ) = z m . In this case p ( STS- 1 ) = ( STS- 1 )( STS- 1 )( STS- 1 ) ... ( STS- 1 )( STS- 1 ) Rearranging the parenthesis we get the pairs S 1 S cancel, except for the S on the left and the S- 1 on the right. We get p ( STS- 1 ) = STTT ...TS- 1 = ST m S- 1 ....
View Full Document

Page1 / 3

hw6sol - Winter 2011 Math 67 Linear Algebra Homework 6...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online