You are about to erase your work on this activity. Are you sure you want to do this?
Updated Version Available
There is an updated version of this activity. If you update to the most recent version of this activity, then your current progress on this activity will be erased. Regardless, your record of completion will remain. How would you like to proceed?
Mathematical Expression Editor
[?]
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 ?
Remember that a linearly independent set can be extended to form a basis.
Click the arrow to see the answer.
Let be a basis for Then there is a basis for and which are respectively It follows
that you must have and so you must have
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 .
Choose a basis of and extend it to a basis
of . Show that is a basis of .