Find a subset of S that is a basis for W

[Guest]

Hi

Let W be the span of S = [3 -2 1], [2 -1 1], [-8 5 -3], [-4 2 -2]

Find a subset of S that is a basis for W

The answer is (3 -2 1) (2 -1 1) (these are in vector form)

I put the vectors of S into a matrix and did reduced echelon form and came out with
[1 0 -2 0]
[0 1 -1 -2]
[0 0 0 0]

but I don't get the same thing as (3 -2 1) (2 -1 1)

I was just wondering why (3 -2 1) (2 -1 1) is the answer?

Thanks for all of your wonderful help
Take care,
Beckie

 

[galactus]

If you perform a reduced echelon form.

The row vectors with the leading 1's form a basis for the row space, which in this case is [3, -2, 1] and [2, -1, 1]

\(\displaystyle \begin{array}{cc}1&-5/8&3/8\\0&1&1\\0&0&0\\0&0&0\end{array}\)

 

[pka]

If you are going to use row reduction, write the vector as row elements
\(\displaystyle \left[ \begin{array}{rrr}
3 & - 2 & 1 \\
2 & - 1 & 1 \\
- 8 & 5 & - 3 \\
- 4 & 5 & - 2 \\
\end{array} \right]\), the rref is \(\displaystyle \left[ \begin{array}{rrr}
1 & 0 & 1 \\
0 & 1 & 1 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{array} \right].\)

Now we can see that the third and forth vectors are dependent on the first two.

 

[Guest]

Thank you