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Let T: R3 -> M(2,2) be the linear transformation given by

T(x,y,z) = [ z .......-z ]

..............[ 0 ..... x-y]

Fix bases B = {(1,0,0),(0,1,0),(0,0,1)} and C = { [1 0] , [0 1] , [0 0] , [0 0] }

..................................................................[0 0].....[0 0]...[1 0]...[0 1]

for R3 and M(2,2) respectively

a) Find the matrix [T]c,b of T with respect to the bases B and C.

b) Use the matrix from part (a) to find the basis for Im(T)

c) Use the matrix from part (a) to find the basis for Ker(T)

For parts b and c I'm pretty sure thats just finding the column space (part b) and solution space (part c) of the matrix.

As for a its more so to do with my own confusion when it comes to bases.

I worked out the matrix to be [ 0 0 0 1 ]

.........(3 by 4 matrix)...........[ 0 0 0 -1]

........................................[ 1 -1 0 0]

The way i got this was by taking each vector from B applying the transformation then writing it in terms of the basis C and then writing the coefficients of each matrix in the rows of the above matrix. I hope that even makes sense >_>.

Any help greatly appreciated :)

P.S. sorry for the garbagety matrices :S

EDIT: Maybe its the transpose of that matrix.... im lost >.<

T(x,y,z) = [ z .......-z ]

..............[ 0 ..... x-y]

Fix bases B = {(1,0,0),(0,1,0),(0,0,1)} and C = { [1 0] , [0 1] , [0 0] , [0 0] }

..................................................................[0 0].....[0 0]...[1 0]...[0 1]

for R3 and M(2,2) respectively

a) Find the matrix [T]c,b of T with respect to the bases B and C.

b) Use the matrix from part (a) to find the basis for Im(T)

c) Use the matrix from part (a) to find the basis for Ker(T)

For parts b and c I'm pretty sure thats just finding the column space (part b) and solution space (part c) of the matrix.

As for a its more so to do with my own confusion when it comes to bases.

I worked out the matrix to be [ 0 0 0 1 ]

.........(3 by 4 matrix)...........[ 0 0 0 -1]

........................................[ 1 -1 0 0]

The way i got this was by taking each vector from B applying the transformation then writing it in terms of the basis C and then writing the coefficients of each matrix in the rows of the above matrix. I hope that even makes sense >_>.

Any help greatly appreciated :)

P.S. sorry for the garbagety matrices :S

EDIT: Maybe its the transpose of that matrix.... im lost >.<

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