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 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