1. ## 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}.

2. ## Linearly 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

3. Originally Posted by Gamma
all of which are not zero...
You mean "some of which..."?

4. ## 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.

5. ## thanks

thank you so much!