# linearly dependent

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• Apr 19th 2009, 03:41 PM
amitmeh
linearly 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, 04:44 PM
Gamma
Linearly Independent
A set ${x_1, x_2, ..., x_n}$ is said to be linearly independent by definition if $a_1 x_1 + a_2 x_2 + ... + a_n x_n=0$ implies $a_i=0$ for all i.

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

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

QED
• Apr 20th 2009, 12:18 AM
scorpion007
Quote:

Originally Posted by Gamma
all of which are not zero...

You mean "some of which..."?
• Apr 20th 2009, 07:54 AM
Gamma
Yeah
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, 06:35 AM
amitmeh
thanks
thank you so much!