Challenge Problems for Chapter 5

Suppose has dimension and has dimension and they are each contained in a subspace, which has dimension equal to where What are the possibilities for the dimension of ?

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Let be a basis of . Let be an matrix.
(a)
If is invertible, show that is a basis of .
(b)
If is a basis of , show that is invertible.
Show that for any real matrix .
Let be an matrix of rank . Show that .
Let and be subspaces of . If , and , show that .

Bibliography

These problems came from Chapter 6 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)

W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, pp. 358–359, and p. 370.

2024-09-06 23:47:04