Find a basis for the subspace S of R4 consisting of all vect

Find a basis for the subspace S of R^4 consisting of all vectors of the form (a - 2b + c, b - c, c, b + a)^T, where a, b, and c range over all real numbers. Enter your answer as a list of vectors separated by commas, and remember that a vector is entered using angular brackets e.g, (1, 2, 3)^T would be entered as . What is the dimension of S?

Solution

Let A =

1

0

0

1

-2

1

0

1

1

4

1

0

We will reduce A to its RREF as under:

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

Add -1 times the 1st row to the 3rd row

Add 1 times the 2nd row to the 3rd row

Then the RREF of A is

1

0

0

1

0

1

0

3

0

0

1

2

Thus, b+a = (a-2b+c) +3(b-c)+3c. Then ( a-2b+c, b-c, c, b+a) = (a-2b+c,b-c,c, (a-2b+c) +3(b-c)+3c)= (a-2b+c)( 1,0,0,1) + (b-c)( 0,1,0,3)+ c(0,0,1,3). Thus, a basis for S is <( 1,0,0,1),( 0,1,0,3),(0,0,1,3)>. The dimension of S is 3.

1	0	0	1
-2	1	0	1
1	4	1	0