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2 years ago

Let T : V → W be a linear transformation from vector space V into vector space W.

Mathematics

1 answer:

wel2 years ago

7 0

Explanation:

Since {v1,...,vp} is linearly dependent, there exist scalars a1,...,ap, with not all of them being 0 such that a1v1+a2v2+...+apvp = 0. Using the linearity of T we have that

a1*T(v1)+a2*T(v1) + ... + ap*T(vp) = T(a1v19+T(a2v2)+...+T(avp) = T(a1v1+a2v2+...+apvp) = T(0) = 0.

Since at least one ai is different from 0, we obtain a non trivial linear combination that eliminates T(v1) , ..., T(vp). That proves that {T(v1) , ..., T(vp)} is a linearly dependent set of W.

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The graph in the attached figure

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% left side templates
\begin{array}{llll}
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\\ \quad \\
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\end{array}\\\\
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\bullet \textit{ flips it upside-down if }{{ A}}\textit{ is negative}
\\\\
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