Question

Question: Assume $I_F+I_F\ne 0$. If $\left\{u,v,w\right\}$is a basis of V over F, prove that the set$\left\{u+v,u+w,u+w\right\}$ is also a basis.

Short answer

Answer

Hence, $\left\{u+v,u+w,u+w\right\}$is also basis for V.

Step-by-step solution

Explanation

Let$\left\{u,v,w\right\}$ is a basis of over .

i.e$\dim \left(v\left(F\right)\right)=3$

we have to show that ,

$B=\left\{u+v,v+w,u+w\right\}$is also a basis of V.

Since$\left\{u,v,w\right\}$ from a basis for V.

we need to only show that B is linearly independent to be a basis V.

Solution

Let$a\left(u+v\right)+b\left(v+w\right)+c\left(u+w\right)=0$

$\Rightarrow \left(a,c\right)u+\left(b+a\right)v+\left(b+c\right)w=0$

Since$\left\{u,v,w\right\}$ forms a basis for .

$\therefore a+c=0$________(1)

$a+b=0$___________(2)

$b+c=0$___________(3)

Now (1) – (3) gives

$a-b=0$________(4)

(2) – (4) ,gives

2a= 0

a = 0

From (4)

b = 0

Frome (3)

c = 0

Thus, we get a= 0, b = 0 , and c =0

Hence, B is linearly independent and B from a basis for V.