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Q14E

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Found in: Page 437

### 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

# Questions: Let $${F_{\bf{1}}}$$ and $${F_{\bf{2}}}$$ be 4-dimensional flats in $${\mathbb{R}^{\bf{6}}}$$, and suppose that $${F_{\bf{1}}} \cap {F_{\bf{2}}} \ne \phi$$. What are the possible dimension of $${F_{\bf{1}}} \cap {F_{\bf{2}}}$$?

The dimension D of the solution set is, $$2 \le D \le 4$$.

See the step by step solution

## Step 1: Find the dimension of $${F_{\bf{1}}} \cap {F_{\bf{2}}}$$ for complete overlap

$${F_1}$$ and $${F_2}$$ have complete overlap, then $${F_1}$$, $${F_2}$$ have four dimensions.

## Step 2: Find the range of dimension

As there is six-dimensional space, at the minimum, they have $$\left( {6 - 4 = 2} \right)$$ dimensions. So, the range of dimensions is shown below:

$$2 \le D \le 4$$

Thus, the dimension of the solution set is from 2 to 4.

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