Answer to Question #323690 in Linear Algebra for Talifhani

Question

Let S = {w1,w2,...,wk} be a basis for the vector space V. Prove that every vector in V can

be expressed as a linear combination of vectors w1,w2,...,wk

in exactly one .

Answer 1

Let $v∈V$ .

Suppose

$v=a_{1}w_{1}+a_{2}w_{2}+...+a_{k}w_{k}$ ----->(1)

$v=b_{1}w_{1}+b_{2}w_{2}+...+b_{k}w_{k}$ ----->(2)

Subtracting (2) from (1), we have;

$0=(a_{1}-b_{1})w_{1}+(a_{2}-b_{2})w_{2}+...+(a_{k}-b_{k})w_{k}$

Since S is a basis for V, then we have that

$a_{1}-b_{1}=a_{2}-b_{2}=...=a_{k}-b_{k}=0$

Hence, $a_{1}=b_{1}, a_{2}=b_{2}, ..., a_{k}=b_{k}$