Cons the vectors upsilon1 1 1 0 0 upsilon2 0 1 1 0 upsilon

Cons the vectors upsilon_1 = (1, 1, 0, 0), upsilon_2 = (0, 1, 1, 0), upsilon_3 = (0, 0, 1, 1) and upsilon_4 = (1, 0, 0, 1) in R^4. Determine if {upsilon_1, upsilon_2, upsilon_3, upsilon_4} are linearly independent or linearly dependent. If they are linearly dependent, find the nontrivial liner relation amongst them. b) Is span ({upsilon_1, upsilon_2, upsilon_3, upsilon_4}) = R^4. If so, explain why. If not, given an example of a vector omega that is not in span ({upsilon_1, upsilon_2, upsilon_3, upsilon_4}).

Solution

2. (a) Let A =

1

0

0

1

1

1

0

0

0

1

1

0

0

0

1

1

We will reduce A to its RREF as under:

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

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

Add -1 times the 3rd row to the 4th row

Then the RREF of A is

1

0

0

1

0

1

0

-1

0

0

1

1

0

0

0

0

Apparently, v₁ , v₂ , v₃, v₄ are not linearly independent and v₄ = v₁ –v₂ +v₃ or, v₁ –v₂ +v₃-v₄ = 0

(b) It is apparent from the RREF of A that span{ v₁ , v₂ , v₃, v₄ } R⁴ as the last row of the RREF of A is a zero row. The vector, (0,0,0,1)^T is not in span{ v₁ , v₂ , v₃, v₄ }.

1	0	0	1
1	1	0	0
0	1	1	0
0	0	1	1