Linear Algebra Please show all work steps and calculations T

Linear Algebra

Please show all work steps and calculations. Thank you.

Solution

3. T: R³?R² is defined by T(x,y,z) = (x+y,y+z).

(a). We have T(e₁) = T(1,0,0) = (1,0) , T(e₂) = T(0,1,0) = (1,1) and T(e₃) = T(0,0,1) = (0,1). Hence [T] = A=             [ T(e₁) , T(e₁) , T(e₁) ] =   

1

1

0

0

1

1

(b).Ker(T) is the set of solutions to the equation AX = 0. To solve this equation, we have to reduce A to its RREF as under:

Add -1 times the 2nd row to the 1st row

Then the RREf of A is

1

0

-1

0

1

1

Now, if X = (x,y,z)^T, then the equation AX = 0 is equivalent to x-z = 0 or, x = z and y+z = 0 or, y = -z. Then X = (z,-z,z)^T= z(1,-1,1)^T.Hence, {(1,-1,1)^T } is a basis for ker(T).