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Exercises 31 and 32 reveal an important connection between linear independence and linear transformations and provide practice using the definition of linear dependence. Let $V$ and $W$ bevector spaces, let $T : V \rightarrow W$ be a linear transformation, and let $\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}\right\}$ be a subset of $V .$Show that if $\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}\right\}$ is linearly dependent in $V,$ thenthe set of images, $\left\{T\left(\mathbf{v}_{1}\right), \ldots, T\left(\mathbf{v}_{p}\right)\right\},$ is linearly dependent in $W .$ This fact shows that if a linear transformation maps a set $\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}\right\}$ onto a linearly independent set $\left\{T\left(\mathbf{v}_{1}\right), \ldots, T\left(\mathbf{v}_{1}\right)\right\},$ then the original set is linearly independent, too (because it cannot be linearly dependent).

So, $T\left(v_{p}\right)$ is linear combination of vectors in $\left\{T\left(v_{1}\right), \ldots, T\left(v_{p-1}\right)\right\} \operatorname{so}\left\{T\left(v_{1}\right), \ldots, T\left(v_{p}\right)\right\}$ is linearly dependent.

Calculus 3

Chapter 4

Vector Spaces

Section 3

Linearly Independent Sets; Bases

Vectors

Johns Hopkins University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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