The Range of a Linear Transformation: A Vector Space Verification

Determine whether the statement is true or false. Justify your answer.The range of a linear transformation is a vector space.

The statement is true

The statement is true. The range of a linear transformation is indeed a vector space.

To justify this answer, let’s first define a linear transformation. A linear transformation is a function that preserves vector addition and scalar multiplication. In other words, for a linear transformation T and vectors u and v, and scalar c, the following properties hold:

1. T(u + v) = T(u) + T(v)
2. T(cu) = cT(u)

Now, let’s consider the range of a linear transformation T. The range, often denoted as R(T), is the set of all possible outputs (or images) of T. In other words, R(T) is the set of all vectors in the codomain of T that can be obtained by applying T to vectors in the domain.

To show that R(T) is a vector space, we need to demonstrate three key properties of a vector space:

1. Closure under vector addition: For any two vectors u and v in R(T), their sum u + v must also be in R(T).

To prove this, let’s consider two vectors u and v in R(T). By definition of R(T), there exist vectors x and y in the domain of T such that T(x) = u and T(y) = v. Now, using Property 1 of linear transformations, we have:

T(x) + T(y) = T(x + y)

Thus, T(x + y) is in R(T), which means u + v is in R(T), satisfying the closure property.

2. Closure under scalar multiplication: For any vector u in R(T) and any scalar c, the scalar multiple cu must also be in R(T).

To prove this, let u be a vector in R(T) such that there exists a vector x in the domain of T such that T(x) = u. Using Property 2 of linear transformations, we have:

cT(x) = T(cx)

Thus, T(cx) is in R(T), which means cu is in R(T), satisfying the closure property.

3. Contains the zero vector: The vector space R(T) must contain the zero vector, which is the output of T when applied to the zero vector in the domain.

To prove this, let’s consider the zero vector 0 in the domain of T. Using Property 2 of linear transformations, we have:

T(0) = 0

Thus, the zero vector is in R(T), satisfying the property of containing the zero vector.

Since the range of a linear transformation satisfies all three properties of a vector space, we can conclude that the statement “the range of a linear transformation is a vector space” is indeed true.

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