Linear Algebra Let V and W be vector spaces and let T: v --> W be a linear trans

Question

Linear Algebra

Let V and W be vector spaces and let T: v --> W be a linear transformation. Assume that T is one-to-one. Prove that if {v1,...,vp} is a linearly independent subset of V, then {T(v1),...T(vp)} is a linearly independent subset of W.

Hint #1: Use the following definition of one-to-one: A transfromation T is one-to-one if T(u) = t(v) always implies u = v.

Hint #2: THe contrapositive of the statement "If X, then Y" is "If not Y, then not X." A statement and its contrapositive are logically equivalent, so proving one is as good as proving the other. Consider attempting a direct proof of the contrapositive in this case.

Explanation / Answer

Let, a1,....,ap so that:

a1T(v1)+.....+apT(vp)=0

Since T is linear

T(a1v1+...+apvp)=0

Since T is one to one

Hence, a1v1+.....+apvp=0

But v1,...,vp are linearly independent. So, a1=...=ap=0

Hence, {T(v1),...,T(vp)} forms a linearly independent subset of W

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