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6. Suppose T: V → W is a linear transformation. Prove that R(T) is a subspace of W .

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Final answer:

The range R(T) of a linear transformation is shown to be a subspace of W, as it is non-empty, closed under addition, and closed under scalar multiplication, satisfying all the requisite properties of a subspace.

Step-by-step explanation:

Proof that R(T) is a Subspace of W

In order to prove that the range R(T) of a linear transformation T from a vector space V to another vector space W is a subspace of W, we must show two things: firstly that R(T) is non-empty; and secondly, that R(T) is closed under vector addition and scalar multiplication.

1. R(T) is non-empty since by definition of a linear transformation, T(0) = 0 and 0 ∈ W, so 0 ∈ R(T).

2. To show R(T) is closed under addition, let u and v be any two elements in R(T). Then there exist vectors a and b in V such that T(a) = u and T(b) = v. Since T is a linear transformation, T(a + b) = T(a) + T(b) = u + v. Hence, u + v is in R(T).

3. To show R(T) is closed under scalar multiplication, let u be an element in R(T) and c be a scalar. There exists a vector a in V such that T(a) = u. By the linearity of T, T(ca) = cT(a) = cu, which means cu is in R(T).

Based on the above, R(T) satisfies all the properties of a subspace and hence is a subspace of W.

User Valeriy Savchenko
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