Suppose that $T: R^n \rightarrow R^m$ is linear and one-to-one. Let $\{\vec{v}_1, \vec{v}_2, \cdots, \vec{v}_k\}$ be a linearly independent subset of $R^n$
a) Prove that the set $\{T(\vec{v}_1), T(\vec{v}_2), \cdots, T(\vec{v}_k)\}$ is a linearly independent subset of $R^m$.
b) Show by example that (a) is false if $T$ is not one-to-one.
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Pablo
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2What did you try? – xyz Nov 22 '22 at 23:18
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@TheoBendit your link directs here – Anne Bauval Nov 22 '22 at 23:59
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Does this answer your question? Prove Transformation is one-to-one iff it carries linearly independent subsets of $V$ onto Lin. Ind. subsets of $W$. – Anne Bauval Nov 23 '22 at 00:01
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Fill in the details in the following proof.
Proof: Let $a_1,\dots,a_k$ be scalars such that $$a_1Tv_1+\dots+a_kTv_k=0$$ Now use the linearity of $T$ and use the theorem that if $T$ is a one-to-one linear map, then null$(T)=\{0\}$ to conclude that $Tv_1,\dots,Tv_k$ is linearly independent.
The question in part $b$ is wrong as it is not true. Consider the linear map $T : R^3\to R^3$ defined by $$T(x, y, z)=(x, y, 0)$$ Now see for yourself what happens to the linearly independent set $\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$ after you apply the above defined linear map to it.
Seeker
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