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Back to Range Space and Null Space

Exercise 24

Avg difficulty 3.0

Show that for any space VV of dimension nn, the dual space

L(V,R)={h ⁣:V→R∣h is linear}\mathop{{\cal L}}(V,\mathbb{R})=\text{$\{\hskip 1.00006pt\text{$h\colon V\to\mathbb{R}$}\mid\text{$h$ is linear}\hskip 1.00006pt\}$}

is isomorphic to Rn\mathbb{R}^{n}. It is often denoted V∗V^{\ast}. Conclude that V∗≅VV^{\ast}\cong V.

Hefferon, Linear Algebra, Maps Between Spaces, Range Space and Null Space, Exercise 24

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Adapted from Linear Algebra © 2016 Jim Hefferon. Licensed under CC BY-SA 3.0 US. Source: Linear Algebra. Imported at revision df2262e.