Question

Let V and W be n-dimensional vector spaces, and let T:V→W be a linear transformation. Suppose that β is a basis for V. Prove that T is an isomorphism if and only if _____ is a basis for W.

a) T(V)
b) dim(W)
c) β(T)
d) T(β)

Answer 1

To prove that T is an isomorphism if and only if T(V) is a basis for W, we need to show two things. First, if T(V) is a basis for W, then T is an isomorphism. Second, if T is an isomorphism, then T(V) is a basis for W.

Step-by-step explanation:

To prove that T is an isomorphism if and only if T(V) is a basis for W, we need to show two things:

1. If T(V) is a basis for W, then T is an isomorphism.
2. If T is an isomorphism, then T(V) is a basis for W.

1. If T(V) is a basis for W, then T is an isomorphism:

If T(V) is a basis for W, it means that the vectors in T(V) span W and are linearly independent. This implies that T is onto (or surjective) because every vector in W can be expressed as a linear combination of vectors in T(V). It also implies that T is one-to-one (or injective) because if T(u) = T(v), then T(u - v) = 0, which implies that u - v is in the null space of T. Since T is one-to-one, the null space of T contains only the zero vector, which means that u = v. Therefore, T is an isomorphism.

2. If T is an isomorphism, then T(V) is a basis for W:

If T is an isomorphism, it means that T is onto (or surjective) and one-to-one (or injective). This implies that the vectors in T(V) span W because every vector in W can be expressed as a linear combination of vectors in T(V). It also implies that the vectors in T(V) are linearly independent because if T(v_1) = T(v_2) for some v_1 and v_2 in V, then v_1 = v_2 by the injectivity of T, which implies that the coefficients of v_1 and v_2 in the linear combination are all zero. Therefore, T(V) is a basis for W.