Determine if the following statements are true or false If t

Determine if the following statements are true or false. If true, justify your answer. If false give a counterexample. For all parts, V is a vector space over F. (a) if the lists (v_1, v_2), (v_2,V_3), and (v_1, v_3) are linearly independent in V, (b) if v_1, v_n-1 is linearly dependent in V, then v_n span (v_1, v_n-1) (c) if the lists (v_1,V_2) and (w_1,W_2) span V, then either v_1 = w_1 and V_1 = w_1 and v_2 = w_2 or v_1 = w_2) and v_2 = w_1 (d) If v_1, v_2 is a linearly independent list in V, then V_1+V_2, V_1-V_2 is linearly independent list in V

Solution

(a)

False

Let, v1=(1,0),v2=(0,1),v3=(1,1)

vi s are pairwise linearly independent but:

v1+v2=v3

Hence, v1,v2,v3 are linearly dependent.

(b)

False.

Let,

v1=(1,0),v2=(2,0),v3=(0,1)

v1,v2,v3 is a linearly dependent

But v3 does not belong to span{v1,v2}

(c)

False

Let: v1=(1,0),v2=(0,1) which span: R2

w1=(1,1),w2=(2,1)

w2-w1=(1,0)

w1-(w2-w1)=(0,1)

Hence, w1,w2 span R2

(d)

True

Let, a ,b so that:

a(v1+v2)+b(v1-v2)=0

(a+b)v1+(a-b)v2=0

v1,v2 are linearly independent. Hence, a+b=0,a-b=0

a=b=0

Hence, v1+v2,v1-v2 are linearly independent.