3 Suppose V is finite-dimensional and TE L(V). Prove that the following are equivalent. (a) T is invertible. n (b) Tv₁,..., Tv is a basis of V for every basis v₁, Un of V. n (c) Tv₁, ..., Tv is a basis of V for some basis v₁, , Un of V.
3 Suppose V is finite-dimensional and TE L(V). Prove that the following are equivalent. (a) T is invertible. n (b) Tv₁,..., Tv is a basis of V for every basis v₁, Un of V. n (c) Tv₁, ..., Tv is a basis of V for some basis v₁, , Un of V.
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.5: Basis And Dimension
Problem 69E: Find a basis for R2 that includes the vector (2,2).
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Transcribed Image Text:3 Suppose V is finite-dimensional and TE L(V). Prove that the following
are equivalent.
(a) T is invertible.
n
(b) Tv₁,..., Tv is a basis of V for every basis v₁,
Un
of V.
n
(c) Tv₁, ..., Tv is a basis of V for some basis v₁,
, Un
of V.
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