Prove that t is one to one if and only if t carries linearly independent subsets of v onto linearly independent subsets of w.

Answer with Step-by-step explanation:

Suppose T is one-one

Let S be a linearly independent subset of V

We want to show that T(S) is linearly independent.

Suppose T(S) is linearly dependent.

Then there exist and some not all zero scalars such that

T is linear therefore,

T is one-one therefore

N(T)=0

S is linearly independent therefore,

It is contradiction.Hence, T(S) is linearly independent.

Conversely, Suppose that T carries linearly independent subset of V onto linearly independent subsets of W.

Assume that T(x)=0 if the set x is linearly independent

Then, by assumption we conclude that  {0} is  linearly independent but {0} is linearly dependent.

It is contradiction .Hence, the set {x} is linearly dependent which implies that x=0

It means N(T)={0}.Therefore, T is one- one

n

step-by-step explanation:

the correct answer is x/3+10≥32

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