Consider the sets S1 2 1 3 5 2 1 and S2 3 3 4 5 11 18 Neit

Consider the sets S_1 = {(2, -1, -3), (5, 2, 1)} and S2 = {(-3, -3. -4), (-5, -11, -18)}. Neither set is a basis for R3; however, each spans its own subspace of R3. Show that these sets span the same subspace.

Solution

Let A be the matrix with the vectors in S₁ as columns, i.e. let A =

2

5

-1

2

-3

1

The RREF of A is

1

0

0

1

0

0

Further,let B be the matrix with the vectors in S₂ as columns, i.e. let B =

-3

-5

-3

-11

-4

-18

The RREF of B is

1

0

0

1

0

0

Thus, S₂ does not span R³ as the vector (0,0,x) in R³ cannotbe expressed as a linear combination of the vecors in S₂.

Also, A and B have the same RREF. This implies that a common basis for the spaces spanned by both S₁ and S₂ is { (1,0,0),(0,1,0)}. Hence, S₁ and S₂ span the same subspace.

2	5
-1	2
-3	1