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Suppose vectors $\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}$ span $\mathbb{R}^{n},$ and let $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a linear transformation. Suppose $T\left(\mathbf{v}_{i}\right)=\mathbf{0}$ for $i=1, \ldots, p$ Show that $T$ is the zero transformation. That is, show that if $\mathbf{x}$ is any vector in $\mathbb{R}^{n},$ then $T(\mathbf{x})=\mathbf{0} .$

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Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 8

Introduction to Linear Transformations

Introduction to Matrices

Missouri State University

University of Michigan - Ann Arbor

Lectures

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So we're given that the vectors v one v p. Span are and tease alone your transformation on We want to show that teest zero transformation. So, um, if X is any vector in on our end t of X is equal to zero. So if X is been any vector on our end and all the vector of these span our end, we can write our vector X as a linear combination of the V's. So, in other words, um, just for some Constance C, we can write X equals C one V one plus C to be to all these reserve vectors plus dot, dot dot Ah, C p VP. So now let's take t of both sides. T of X is equal to t of C one V one plus up to C P V P Tease Lanier s so we can pull Constance out. So we have C one t of V one close that dot dot c P t of v p, and were given that each individual t of the eye is equal to zero for I goes from one to pee. So from star, which is ah this given Here we have C one time zero plus c two times zero plus all the way up to C P Times zero, which is just equal to zero. So therefore t of any vector X is equal.

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