Let Y Vrightarrow 1 Vrightarrow 2 subset R3 where Vrightarr

Let Y = {V^rightarrow _1, V^rightarrow _2} subset |R^3, where V^rightarrow _1 = emacr _1 + emacr _2, and V^rightarrow _2 = emacr _2 + emacr _3. Show that Y is linearly independent Find a vector V^rightarrow _3 which is not in the sapn of y Let z = {V_1, V_2, V_3}. I z a basis for |R^3?

Solution

a)Let a,b so that

av1+bv2=0

a(e1+e2)+b(e2+e3)=0

ae1+(a+b)e2+be3=0

HEnce, a=b=0

So the vectors are linearly independent

b)

Let, v3=e1

Let, a,b so that

v3=e1=a(e1+e2)+b(e2+e3)=ae1+(a+b)e2+be3

So, a=1,b=0

a+b=0

Hence no solution

SO, v3=e1 is not in span

c)

Yes.

R3 has dimension 3 so any set of 3 linearly independent vectors is a basis for R3