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Call a subset \(S\) of a vector space \(V\) a spanning set if \(\operatorname{Span}(S)=V\). Suppose that \(T: V \rightarrow W\) is a linear map of vector spaces.

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Call a subset \(S\) of a vector space \(V\) a spanning set if \(\operatorname{Span}(S)=V\). Suppose that \(T: V \rightarrow W\) is a linear map of vector spaces.
a) Prove that a linear map \(T\) is 1-1 if and only if \(T\) sends linearly independent sets to linearly independent sets.
b) Prove that \(T\) is onto if and only if \(T\) sends spanning sets to spanning sets.
by ♦ Diamond (89,175 points) | 193 views

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