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About LinkBacksHi 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 setis said to be linearly independent by definition if
implies
for all i.
Supposeare 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
Showing it to be in the span as desired.
Conversely if y is in the span ofbut then we would have
all of which are not zero, so
QED
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Originally Posted by Gamma 
