## A couple of questions...

Linear spaces and subspaces, linear transformations, bases, etc.
isuckatmath
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Joined: Wed Oct 07, 2009 2:22 am
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### A couple of questions...

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.

stapel_eliz
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Joined: Mon Dec 08, 2008 4:22 pm
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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.)
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.
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.

isuckatmath
Posts: 3
Joined: Wed Oct 07, 2009 2:22 am
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### Re: A couple of questions...

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