Question #323690

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 .

1
Expert's answer
2022-04-05T14:34:08-0400

Let vVv∈V .


Suppose

v=a1w1+a2w2+...+akwkv=a_{1}w_{1}+a_{2}w_{2}+...+a_{k}w_{k} ----->(1)

v=b1w1+b2w2+...+bkwkv=b_{1}w_{1}+b_{2}w_{2}+...+b_{k}w_{k} ----->(2)


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


0=(a1b1)w1+(a2b2)w2+...+(akbk)wk0=(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


a1b1=a2b2=...=akbk=0a_{1}-b_{1}=a_{2}-b_{2}=...=a_{k}-b_{k}=0


Hence, a1=b1,a2=b2,...,ak=bka_{1}=b_{1}, a_{2}=b_{2}, ..., a_{k}=b_{k}








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