Challenge Problems for Chapter 9

If is a subspace of , show that .
Show that , then use Theorem th:023783c twice.
If is a subspace of , show how to find an matrix such that .
If and are subspaces, define to be the set of all possible sums of elements of with elements of .
(a)
Is a subspace? Show that .
(b)
Illustrate this result with a diagram in where and are two distinct lines through the origin. What does look like? What do and look like?
A square matrix satisfying is called a projection matrix. If is a projection matrix, show that is also a projection matrix.
Let be an matrix, and let . Show that the following are equivalent.
(a)
( is a projection matrix).
(b)
for all and in .
(c)
for all in .
For (ii) implies (iii): Write and use the uniqueness argument preceding the definition of . For (iii) implies (ii): is in for all in .
If and and are projection matrices, show that is also a projection matrix.
If is and is invertible, show that is a projection matrix.

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Consider where one of . Show that the characteristic polynomial (see Definition def:char_poly_complex) is given by , where , and find an orthogonal matrix such that is diagonal.

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Given , show that the characteristic polynomial (see Definition def:char_poly_complex) is given by , and find an orthogonal matrix such that is diagonal.

Bibliography

Some of these problems come from Section 7.4 of Ken Kuttler’s A First Course in Linear Algebra. (CC-BY)

Ken Kuttler, A First Course in Linear Algebra, Lyryx 2017, Open Edition, pp. 433–438.

Other problems come from the second part of Section 8.1 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)

W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, p. 422–423

2024-09-06 23:47:04