zi2lalZ

2021-01-17

Let $v}_{1},{v}_{2},\dots .,{v}_{k$ be vectors of Rn such that

$v={c}_{1}{v}_{1}+{c}_{2}{v}_{2}+\dots +{c}_{k}{v}_{k}={d}_{1}{v}_{1}+{d}_{2}{v}_{2}+\dots +{d}_{k}{v}_{k}$ .

for some scalars$c}_{1},{c}_{2},\dots .,{c}_{k},{d}_{1},{d}_{2},\dots .,{d}_{k$ .Prove that if $ci\ne dif{\textstyle \phantom{\rule{1em}{0ex}}}\text{or}{\textstyle \phantom{\rule{1em}{0ex}}}somei=1,2,\dots .,k$ ,

then$v}_{1},{v}_{2},\dots .,{v}_{k$ are linearly dependent.

for some scalars

then

Talisha

Skilled2021-01-18Added 93 answers

You haven't mentioned what v is so I'm going to ignore it.

We know that

for some

Now suppose that for some

By supposition, there is a solution to the equation

such that not all of the

But let's say your class defines linearly dependent as meaning that at least one of the vectors is expressible as a linear combination of the others. Then we just move all of the terms except the jth one to the RHS.And here's what we get by saying

Doing so we get