Question

Linear algebra:

Determine the basis of span{ [1,-1,1], [1,0,2], [1,-2,0], [1,1,-1]

Answer 1

After row reduction i got:

1 0 0 -4
0 1 0 7/2
0 0 1 3/2

and now?

Answer 2

um im pretty sure that you can just take th first three vectors and say they are a basis becuase hte 4th vector is just a linear combination of the 3

Answer 3

X1 = 4 X4
X2 = 7/2 X4
X3 = 3/2 X4
X4 = free

X4(4,7/2,3/2,1) ?

Answer 4

{[1,-1,1], [1,0,2], [1,-2,0]} spans the same space as
{[1,-1,1], [1,0,2], [1,-2,0], [1,1,-1]} and teh former is linearly independent so its a basis

Answer 5

Or in other words, the four vectors span all of R^3 and hence
{[1,0,0], [0 1 0], [0 0 1]} is also a basis.

Answer 6

so x1, x2 and x3 span the basis

Answer 7

x1, x2, x3 is a basis for the span

Answer 8

x1 |4 |
x = x2 = x4 |7/2|
x3 |3/2|
x4 |1 |

Answer 9

No, that can't be right because if it were, the sub-space that is the span of those vectors would be one dimensional.

How many dimensions does the span of { [1,-1,1], [1,0,2], [1,-2,0], [1,1,-1] } have?

Answer 10

4

Answer 11

Haven't you just shown that the first three vectors are linearly independent? So at least 3.

Now 4 can't be right either because R^3 only has 3 dimensions. And it isn't the case that all four vectors are linearly independent.

Answer 12

owh shoot... i ment 3 because i have 3 pivots

Answer 13

Hence the span is three dimensional and any basis will have three basis vectors.

Answer 14

the vectors [1,-1,1], [1,0,2], [1,-2,0] will do. But as this span is also R^3, the basis
[1 0 0], [0 1 0], [0 0 01] also works.

Answer 15

ahh oke.. thanks for the explanation :)