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Found in: Page 191

### Linear Algebra and its Applications

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.

See the step by step solution

## Step 1: Find the dimension of H and V

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

For $$n = 0$$,

$$H = V = \left\{ 0 \right\}$$.

## Step 2: Check for $$H = V$$

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.