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Chapter 7: Q66E (page 325)

(a) Give an example of a 3 脳 3 matrix A with as many nonzero entries as possible such that both span(e鈫�1) and span(e鈫�1,e鈫�2) are A-invariant subspaces of . See Exercise 65.

(b) Consider the linear space Vof all 3 脳 3 matrices A such that both span (e鈫�1) and span (e鈫�1,e鈫�2) are A-invariant subspaces of R3 . Describe the space V (the matrices in V 鈥渉ave a name鈥�), and determine the dimension of V.

Short Answer

Expert verified

The solutions are,

(a)A=2711024143700-3.11

(b) those are upper triangular matrices, and dim V = 6

Step by step solution

01

Solving for (a)

Since span (e鈫�1) isA-invariant, the first column of the matrix is;
a00

Since span (e鈫�1,e鈫�2) is also A-invariant, then the second column must be

bc0

The third column is clearly arbitrary.

Thus, an example of such a matrix is:

A=2711024143700-3.11

02

Solving for (b)

The difference from part a. is that these matrices do not necessarily have as many non-zero entries as possible. Still, as seen also from part a, these matrices are in form

A=abc0de00f

These are upper triangular matrices. It applies dim V=6

Hence, (a)A=2711024143700-3.11

(b) those are upper triangular matrices, and dim V = 6.

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