Hello I do not indestand how I can solve this one, can someone give me a hint please. Thank you.

the answer in the book is

1 0

0 1

-1 0

Find a subset of S with the same span as S that is as small as possible.

1 -2 0

0 0 1

-1 2 0

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- Nov 10th 2012, 02:20 PMmathproblemsSpan as small as possible
Hello I do not indestand how I can solve this one, can someone give me a hint please. Thank you.

the answer in the book is

1 0

0 1

-1 0

Find a subset of S with the same span as S that is as small as possible.

1 -2 0

0 0 1

-1 2 0 - Nov 10th 2012, 02:37 PMfkfRe: Span as small as possible
Column 2 is just a multiple of column one. Hence we can remove that vector (column 2) in the set S, and still have the same span, but with a smaller set.

- Nov 10th 2012, 02:42 PMPlatoRe: Span as small as possible
- Nov 10th 2012, 02:54 PMmathproblemsRe: Span as small as possible
thank you!

and the next problem is simular.

1 -2 0

-2 4 0

1 -2 0

it would be just

1

-2

1

because

-2 *1 = -2

-2*-2 = 4

-2*1 = -2

and also would be a reduced row echalon form need here?

1 -2 0

0 0 0

0 0 0

This is the way to find the -2 term in the correct way? - Nov 10th 2012, 03:02 PMmathproblemsRe: Span as small as possible
and one more:

-1 0 1

0 1 2

1 2 3

Reduced row echalon form will be

1 0 -1

0 1 2

0 0 0

I still do not get this one... - Nov 11th 2012, 12:38 AMfkfRe: Span as small as possible
The 0 vector doesn't add anything new to the span since we can have the weight 0 on the first vector, and still get the 0 vector. Hence we can remove that and have the same span. The second column vector is still a multiple of the first one, and therefore not adding any new to the span so we can remove this one also, and hence only have the first column vector. Still having the same span as we had when we had the set S = {v1,v2,v3}

The row echelon form is correct, and there you clearly we that the second column is just a multiple of the first column. - Nov 11th 2012, 12:40 AMfkfRe: Span as small as possible
- Nov 11th 2012, 01:30 PMmathproblemsRe: Span as small as possible
Not sure, how this can be reduced again?

1 0 -1

0 1 2

0 0 0 - Nov 11th 2012, 01:39 PMmathproblemsRe: Span as small as possible
and one more:

-1 0 1

0 1 2

1 2 3

Reduced row echalon form will be

1 0 -1

0 1 2

0 0 0

the answer in the book is

-1 0

0 1

1 2

I still do not get this one...is it because the last column is

-1

2

0 and it it bigger than other two

1 0

0 1

0 0 - Nov 12th 2012, 01:48 AMfkfRe: Span as small as possible
- Nov 12th 2012, 09:46 AMmathproblemsRe: Span as small as possible
thank you so much.