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}.
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}.
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