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}.

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- Apr 19th 2009, 02:41 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}. - Apr 19th 2009, 03:44 PMGammaLinearly Independent
A set $\displaystyle {x_1, x_2, ..., x_n}$ is said to be linearly independent by definition if $\displaystyle a_1 x_1 + a_2 x_2 + ... + a_n x_n=0$ implies $\displaystyle a_i=0$ for all i.

Suppose $\displaystyle {x_1, x_2,... x_n, y}$ are linearly dependent, then there exist coefficients, not all of which are 0, such that $\displaystyle a_1 x_1 + a_2 x_2 + ... + a_n x_n + a_{n+1} y=0 $. Notice that if $\displaystyle a_{n+1} = 0$ then we would contradict the linear independence of $\displaystyle {x_1, x_2, ..., x_n}$. Thus we can say $\displaystyle y=-(a_2 x_2 + ... + a_n x_n)/a_{n+1}$ Showing it to be in the span as desired.

Conversely if y is in the span of $\displaystyle {x_1, x_2, ..., x_n}$ then there exist coefficients such that $\displaystyle y=a_1 x_1 + a_2 x_2 + ... + a_n x_n$ but then we would have$\displaystyle 0=a_1 x_1 + a_2 x_2 + ... + a_n x_n + (-1)y$ all of which are not zero, so $\displaystyle {x_1, x_2,... x_n, y}$ is linearly dependent.

QED - Apr 19th 2009, 11:18 PMscorpion007
- Apr 20th 2009, 06:54 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. - Apr 21st 2009, 05:35 AMamitmehthanks
thank you so much!