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.
To justify this answer, let’s start by defining the terms involved. A linear transformation is a function that preserves addition and scalar multiplication. In other words, for any vectors u and v, and any scalar c, if T is a linear transformation, it satisfies the following properties:
1. T(u + v) = T(u) + T(v)
2. T(cu) = cT(u)
The range of a linear transformation, also known as the image, is the set of all possible outputs when applying the transformation to the elements of its domain.
Now, let’s prove that the range of a linear transformation is a vector space.
For a set to be a vector space, it must satisfy the following properties:
1. Closure under addition: For any two vectors u and v in the set, their sum u + v should also be in the set.
2. Closure under scalar multiplication: For any vector u in the set and any scalar c, the product cu should also be in the set.
3. The set must contain the zero vector.
4. The set must be associative under addition.
5. The set must have an additive identity.
6. The set must be associative and distributive under scalar multiplication.
Considering a linear transformation T, let’s examine these properties for its range:
1. Closure under addition: Let’s consider two vectors w1 and w2 in the range of T. By definition, there exist vectors u1 and u2 in the domain of T such that T(u1) = w1 and T(u2) = w2. Since T is a linear transformation, T(u1 + u2) = T(u1) + T(u2) = w1 + w2. Therefore, the sum w1 + w2 is also in the range of T, proving closure under addition.
2. Closure under scalar multiplication: Let w be a vector in the range of T, so there exists a vector u in the domain of T such that T(u) = w. For any scalar c, T(cu) = cT(u) = cw. Therefore, the scalar multiple cw is also in the range of T, proving closure under scalar multiplication.
3. The zero vector: Since T is a linear transformation, it maps the zero vector in the domain to the zero vector in the range. Therefore, the zero vector is always in the range of T.
4. Associativity under addition: This property holds for vector addition in a vector space, which is inherited by the range of T by the closure under addition property.
5. Additive identity: This property holds in a vector space, and it is preserved in the range of T since it contains the zero vector.
6. Associativity and distributivity under scalar multiplication: These properties hold in a vector space, and they are preserved in the range of T since it contains all possible scalar multiples of vectors in the domain.
Since the range of T satisfies all the properties of a vector space, we can conclude that the statement is true: the range of a linear transformation is indeed a vector space.
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