A couple of questions...  TOPIC_SOLVED

Linear spaces and subspaces, linear transformations, bases, etc.

A couple of questions...

Postby isuckatmath on Wed Oct 07, 2009 2:41 am

I am having some trouble with the following two questions. Any help would be greatly appreciated!

The first question:
a. Use matrix multiplication to show that if x0 is a solution of the homogeneous system Ax = 0 and x1 is a solution of the nonhomogeneous system Ax = b, then x0 + x1 is also a solution of the nonhomogeneous system.
b. Suppose that x1 and x2 are solutions of the nonhomogeneous system of part (a). Show that x1 - x2 is a solution of the homogeneous system Ax = 0.

The second question:
Let A be an n x n matrix such that Ax = x for every n-vector x. Show that A = I.
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Postby stapel_eliz on Wed Oct 07, 2009 3:32 pm

isuckatmath wrote:The first question:
a. Use matrix multiplication to show that if x0 is a solution of the homogeneous system Ax = 0 and x1 is a solution of the nonhomogeneous system Ax = b, then x0 + x1 is also a solution of the nonhomogeneous system.

If Ax0 = 0 and Ax1 = b, then what is the result of A(x0 + x1)? (Hint: Multiplication distributes over addition.) :wink:

isuckatmath wrote:b. Suppose that x1 and x2 are solutions of the nonhomogeneous system of part (a). Show that x1 - x2 is a solution of the homogeneous system Ax = 0.

Do the same distribution thing. :D

isuckatmath wrote:The second question:
Let A be an n x n matrix such that Ax = x for every n-vector x. Show that A = I.

Since Ax = x, then Ax - x = Ax - Ix = 0. Now do the reverse of distribution, and see what you get. 8-)
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Re: A couple of questions...  TOPIC_SOLVED

Postby isuckatmath on Wed Oct 07, 2009 9:30 pm

omg that was so much easier than i was making it out to be. Thanks so much!!!!! :D
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