Suppose that are inverses.

If { } is a basis for a subspace of and , prove that { } is a basis for .

In addition, give an example to show that this need not be true if T does not have an inverse.

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- February 2nd 2010, 11:59 AMRuntyProof for a basis of a linear transformation
Suppose that are inverses.

If { } is a basis for a subspace of and , prove that { } is a basis for .

In addition, give an example to show that this need not be true if T does not have an inverse. - February 2nd 2010, 04:37 PMDefunkt
- February 4th 2010, 06:30 AMRunty
- February 4th 2010, 03:04 PMDefunkt
- February 6th 2010, 07:38 PMwopashui
- February 7th 2010, 11:31 AMRunty
- February 7th 2010, 12:14 PMDefunkt
Take any transformation that is not invertible; for example, defined by . Since T is not invertible, . In fact, .

Then, is spanned by , however which is not a base.

This will work for any transformation that is not invertible:

Since it is not invertible, but for any , , and therefore the image of any base of will be mapped to the zero vector. - February 7th 2010, 12:16 PMRunty
- February 7th 2010, 09:27 PMmath2009

is linearly independent

it's basis of V - February 8th 2010, 06:00 AMHallsofIvy
"ker(T)" is the "kernel" of T, also called the "null space" of T. It is the subspace of all vectors, v, such that Tv= 0.

If T is invertible, then Tv= 0 gives or v= 0. That is, if T is invertible, its kernel (null space) consists only of the 0 vector.

In fact, you can also prove the other way: if the 0 vector is the only vector in the kernel of T, T is invertible.

"null space" is used exclusively in linear algebra. "kernel" of an operator is also used in group theory, ring theory, etc.