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Q8SE

Expert-verifiedFound in: Page 191

Book edition
5th

Author(s)
David C. Lay, Steven R. Lay and Judi J. McDonald

Pages
483 pages

ISBN
978-03219822384

**Let H be an n-dimensional subspace of an n-dimensional vector space V. Explain why \(H = V\).**

*H* is a subspace of *V,* and *B* is also in *V*. As *B* is a linearly independent set in *V*, *B* must also be a basis for *V*.

The bases of *H* and *V* have exactly *n* vectors because *H* is an *n*-dimensional subspace of *V*.

For \(n = 0\),

\(H = V = \left\{ 0 \right\}\).

For subspace *H* is *n*-dimensional subspace, there is a *B* for *H*. *B *must have *n* elements and be linearly independent.

Since *H* is a subspace of *V* and *B* is also in *V*, *B* is a linearly independent set in *V*. So, *B* must also be a basis for *V*. Hence, *H *and *V* are the same.

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