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Linear Algebra and its Applications
Found in: Page 437
Linear Algebra and its Applications

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

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Short Answer

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 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.

Most popular questions for Math Textbooks

Let \({v_1} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{1}}\end{array}} \right]\), \({v_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{5}}\end{array}} \right]\), \({v_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{4}}\\{\bf{3}}\end{array}} \right]\), \({p_1} = \left[ {\begin{array}{*{20}{c}}{\bf{3}}\\{\bf{5}}\end{array}} \right]\), \({p_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{5}}\\{\bf{1}}\end{array}} \right]\), \({p_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{3}}\end{array}} \right]\), \({p_{\bf{4}}} = \left[ {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{\bf{0}}\end{array}} \right]\), \({p_{\bf{5}}} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{4}}\end{array}} \right]\), \({p_{\bf{6}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{2}}\end{array}} \right]\), \({p_{\bf{7}}} = \left[ {\begin{array}{*{20}{c}}{\bf{6}}\\{\bf{4}}\end{array}} \right]\) and let \(S = \left\{ {{v_1},{v_2},{v_3}} \right\}\).

  1. Show that the set is affinely independent.
  2. Find the barycentric coordinates of \({p_1}\), \({p_{\bf{2}}}\), and \({p_{\bf{3}}}\) with respect to S.
  3. On graph paper, sketch the triangle \(T\) with vertices \({v_1}\), \({v_{\bf{2}}}\), and \({v_{\bf{3}}}\), extend the sides as in Figure 8, and plot the points \({p_{\bf{4}}}\), \({p_{\bf{5}}}\), \({p_{\bf{6}}}\), and \({p_{\bf{7}}}\). Without calculating the actual values, determine the signs of the barycentric coordinates of points \({p_{\bf{4}}}\), \({p_{\bf{5}}}\), \({p_{\bf{6}}}\), and \({p_{\bf{7}}}\).

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