Finding kernel

[corvetx13]

The following problem is puzzling me so any help would be appreciated!

 

Consider the linear transformation L: M_22 -> R^4 defined by

 

L of

 

(a b

c d)

 

=

 

(-10a - c + 2d, 5a + b - d, 2c + d, 9a + 17b + 2d)

 

Compute ker(L).

 

Okay, so far I've put the matrix:

10 0 -1 2

5 1 0 -1

0 0 2 1

9 17 0 2

 

into row reduced echelon form to get

 

1 0 0 -1/4

0 1 0 1/4

0 0 1 1/2

0 0 0 0

 

And from this, I've gotten that

 

a = 1/4 d

b = -1/4 d

c = -1/2 d

and d is arbitrary

 

But how do I notate ker(L)?

[timo]

how about [math] K = \{ \left( \begin{array}{cc} \frac{1}{4} & \frac{-1}{4} \\ \frac{-1}{2} & 1 \end{array} \right) \, r, \ r \in R \} [/math] ?

Note: I didn´t check your math but I assumed you did calculate the elements of M_22 that map onto 0 of R^4, which I think is the definition of the kernel.