Hi guys, I need help with the following question:

Given that {X1, X2,..., Xk} is a linearly independent set, show that {X1, X2,..., Xk, y} is linearly dependent if and only if y is in the span {X1, X2,..., Xk}.

Printable View

- April 19th 2009, 03:41 PMamitmehlinearly dependent
Hi guys, I need help with the following question:

Given that {X1, X2,..., Xk} is a linearly independent set, show that {X1, X2,..., Xk, y} is linearly dependent if and only if y is in the span {X1, X2,..., Xk}. - April 19th 2009, 04:44 PMGammaLinearly Independent
A set is said to be linearly independent by definition if implies for all i.

Suppose are linearly dependent, then there exist coefficients, not all of which are 0, such that . Notice that if then we would contradict the linear independence of . Thus we can say Showing it to be in the span as desired.

Conversely if y is in the span of then there exist coefficients such that but then we would have all of which are not zero, so is linearly dependent.

QED - April 20th 2009, 12:18 AMscorpion007
- April 20th 2009, 07:54 AMGammaYeah
yeah, i guess I can see the confusion there.

I mean all of them cannot be zero, so yeah some of them are nonzero, same thing. - April 21st 2009, 06:35 AMamitmehthanks
thank you so much!