be defined by

Find the basis for ker L and basis for range L

My work:

--->

Sorry, I don't know how to find the kernel for an Mnn --> Mnn with an expression like this(Doh) Can someone please help me out with this. Thanks in advance

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- Jul 25th 2010, 09:05 PMwilday86Kernel and range of a linear Transformation
be defined by

Find the basis for ker L and basis for range L

My work:

--->

Sorry, I don't know how to find the kernel for an Mnn --> Mnn with an expression like this(Doh) Can someone please help me out with this. Thanks in advance - Jul 25th 2010, 11:57 PMroninpro
You are fine all the way up to here:

The step after this does not make sense. Check your matrix multiplication! - Jul 26th 2010, 07:38 PMwilday86
I apologize. I got pretty sloppy there. I was doing that late at night and so badly wanted to crash. Ok Here is what I got so far. Hopefully there wont typos...

I turned this into a matrix and performed rref

a= 0

b = 2

c = arbitrary

d - 0

However the correct answer for the basis for ker L is:

I don't know how to answer this. (Worried)

I also have to find the basis for range L.

The correct answer for the basis for range L is:

But I don't know how to get that since I don't know how to get the ker L for this problem. I know that range L = span S. Not sure how to set this up....

Can someone please help me figure out this problem. Thanks in advance. - Jul 27th 2010, 05:35 AMHallsofIvy
The lower left entry should be "a- d", not "a+ d".

Rather than go to 5 by 4 matrices, note that this says the 2c- b= 0, 2d- 2a= 0, a- d= 0, and b- 2c= 0. The second and third equations both give a= d. From the first equation b= 2c so the fourth equation becomes 2c- 2c= 0 which is true for all c. These will be true for all d= a and b=2c: .

Quote:

I turned this into a matrix and performed rref

a= 0

b = 2

c = arbitrary

d - 0

However the correct answer for the basis for ker L is:

Quote:

I don't know how to answer this. (Worried)

I also have to find the basis for range L.

The correct answer for the basis for range L is:

But I don't know how to get that since I don't know how to get the ker L for this problem. I know that range L = span S. Not sure how to set this up....

Can someone please help me figure out this problem. Thanks in advance.

The range of L is all matrics [tex]\begin{bmatrix}u & v \\ w & x\end{bmatrix} such that for**some**a, b, c, d. We need to reduce u= 2c- b, v= 2d- 2a, w= a- d, and x= b- 2c to equations in u, v, w, and x only. Obviously the second and third equation say v= 2w. From the first equation b= 2c- u and then the fourth equation becomes x= (2c-u)- 2c= -u. We can write any matrix in the range of L as which is equivalent to the basis you give.