Question

Mark the following as true or false, as the case may be. If a statement is true, then prove it. If a statement is false, then provide a counter-example.

a) A set containing a single vector is linearly independent

b) The set of vectors {v, kv} is linearly dependent for every scalar k

c) every linearly dependent set contains the zero vector

d) The functions f₁ and f₂ are linearly dependent is there is a real number x, so that k₁f₁(x)+k₂f₂(x)=0 for some scalars k₁ and k₂

Answer 1

a)  False.

Take singleton set {0}, which is well known Linearly Dependent

b)  True.

To prove :{v, kv} are linearly dependent.

On, contrary assume it is Linearly Independent.

For, any two arbitrary constant a, b. Consider linear combination,

a.v+ b(kv) =0

so, (a+bk) v=0

so, a+bk=0, { as we take v is non zero vector}

so, a=-bk,   

Constants are non zero, so it is not linearly independent.

Hence, it is linearly dependent.

c) False.

Consider set, {2, 4} is linearly dependent from (b)

But, It donot have zero.

d) False

Take, f1(x) =2 , f2(x) =4, i. e. f2=2f1, so linearly dependent.

take, k1=1,k2=1

So, k1f1+k2f2=f1(x) +f2(x)=2+4=6

So, it will never be zero in this case for any value of x.