If W is a subspace of a vector space V and

killjoy1990xb9
2022-01-04
Answered

Prove that

If W is a subspace of a vector space V and$w}_{1},{w}_{2},\dots ,{w}_{n$ are in W, then ${a}_{1}{w}_{1}+{a}_{2}{w}_{2}+\dots +{a}_{n}{w}_{n}\in W$ for any scalars $a}_{1},{a}_{2},\dots ,{a}_{n$ .

If W is a subspace of a vector space V and

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Linda Birchfield

Answered 2022-01-05
Author has **39** answers

Here W is a subspace of a vector space V and $w}_{1},{w}_{2},\dots ,{w}_{n$ are in W.

From above,${w}_{1}\in W,{w}_{2}\in W,\dots .,{w}_{n}\in W$ .

Since, W is closed under scalar multiplication${a}_{1}{w}_{1}\in W,{a}_{1}{w}_{2}\in W,\dots .,{a}_{1}{w}_{n}\in W$ for any scalars $a}_{1},{a}_{2},\dots ,{a}_{n$ .

W is closed under addition.

Hence,${a}_{1}{w}_{1}+{a}_{2}{w}_{2}+\dots +{a}_{n}{w}_{n}\in W$ for any scalars $a}_{1},{a}_{2},\dots ,{a}_{n$ .

From above,

Since, W is closed under scalar multiplication

W is closed under addition.

Hence,

Suhadolahbb

Answered 2022-01-06
Author has **32** answers

I understand how you proved!

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