Question

If vectors$\overrightarrow{u},\overrightarrow{v}and\overrightarrow{w}$ are in the subspace V of $R^n,$then the vector $2\overrightarrow{u}-3\overrightarrow{v}+4\overrightarrow{w}$must be in V as well.

Short answer

The above statement is true.

If vectors $\overrightarrow{u},\overrightarrow{v}and\overrightarrow{w}$are in the subspace V of $R^n,$then the vector $2\overrightarrow{u}-3\overrightarrow{v}+4\overrightarrow{w}$must be in V as well.

Step-by-step solution

One step test of subspace

Let $R^n$be a vector space over the field K, and let V is any subset of$R^n$then V is a subspace of$R^n$ if and only if-

1. $0\in V$
   
2. $\overrightarrow{u},\overrightarrow{v}\in V,a,b\in F$
   

$\Rightarrow a\overrightarrow{u}+b\overrightarrow{v}\in V$

To show the vector 2u→-3v→+4w→ are in the subspace V.

Since it is given that vectors$\overrightarrow{u},\overrightarrow{v}and\overrightarrow{w}$ are in the subspace V ofdata-custom-editor="chemistry" $R^n,$ then we have

1. $0\in V$
   
2. $\overrightarrow{u},\overrightarrow{v}\in V,a,b\in F$

$\Rightarrow a\overrightarrow{u}+b\overrightarrow{v}\in V$

Thus,$2\overrightarrow{u}-3\overrightarrow{v}\in V.$

And so,$2\overrightarrow{u}-3\overrightarrow{v}+4\overrightarrow{w}\in V.$

i.e. $2\overrightarrow{u}-3\overrightarrow{v}+4\overrightarrow{w}$are in the subspace V as well.

Final Answer

If vectors$\overrightarrow{u},\overrightarrow{v}and\overrightarrow{w}$ are in the subspace V of $R^n,$then the vector $2\overrightarrow{u}-3\overrightarrow{v}+4\overrightarrow{w}$ must be in V as well by the definition of subspace.