Solved: Guided Proof Let {v1, v2, . . . , vn} be a basis

Chapter 6, Problem 83

(choose chapter or problem)

Let \({v_{1}, v_{2}, . . . , v_{n}}\) be a basis for a vector space V. Prove that if a linear transformation \(T: V \rightarrow V\) satisfies \(T(v_{i}) = 0\) for i = 1, 2, . . . , n, then T is the zero transformation.

Getting Started: To prove that T is the zero transformation, you need to show that T(v) = 0 for every vector v in V.

(i) Let v be an arbitrary vector in V such that \(v = c_{1}v_{1} + c_{2}v_{2} + . . . + c_{n}v_{n}\).

(ii) Use the definition and properties of linear transformations to rewrite T(v) as a linear combination of \(T(v_{i})\).

(iii) Use the fact that \(T(v_{i}) = 0\) to conclude that T(v) = 0, making T the zero transformation.

Text Transcription:

{v_1, v_2, . . ., v_n}

T: V rightarrow V

T(v_i) = 0

v = c_1v_1 + c_2v_2 + cdot + c_n v_n

T(v_i)