Question

Suppose V and W are finite-dimensional, T ∈ L(V;W), and there exists v′∈ W0​ such that null T0​=span {v′}. Prove that range T=null T′.

a) rangeT = rangeT′
b) rangeT = nullT′
c) nullT = rangeT′
d) nullT = nullT′

Answer 1

The problem deals with the relationship between the range of a linear transformation T and the null space of its dual transformation T'. The range of T is equal to the null space of T', thus the correct answer is b) range T = null T'.

Step-by-step explanation:

The question provided relates to the concepts from linear algebra involving the relationship between the null space and the range (also known as the image) of linear transformations, denoted by T and T' (the dual transformation). To determine the correct relationship, we must understand several definitions:

• The null space (or nullity) of a transformation T consists of all vectors in V that T maps to the zero vector in W.
• The range (or image) of T consists of all vectors in W that are the images of vectors in V under the transformation T.
• The dual space W0 consists of all linear functionals on W.
• The dual transformation T' maps linear functionals in W0 to linear functionals in V0.
• null T0 refers to the null space of the dual transformation T', which includes functionals that annihilate the range of T.

It's given that null T0 is the span of v', which suggests that the range of T is one-dimensional, and v' annihilates all vectors outside this one-dimensional subspace. Therefore, every functional in null T0 will annihilate the range of T, and thus the range of T consists precisely of those vectors annihilated by null T0, which is the same as null T'.

Therefore, the correct option that describes the relationship between these spaces is b) range T = null T'.