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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q3.2-53E (page 133)

Consider a subspace $V$ of ${鈩�}^{n}$ . We define the orthogonal complement $V^{鈯�}$ of $V$ as the set of those vectors $\overset{鈫�}{w}$ in ${鈩�}^{n}$ that are perpendicular to all vectors in ; that is, $\overset{鈫�}{w} 路 \overset{鈫�}{v} = 0$ , for all $\overset{鈫�}{v}$ in . Show that is a subspace of ${鈩�}^{n}$ .

Short Answer

Expert verified

The orthogonal complement, $V^{鈯�}$ of $V$ is a subspace of ${鈩�}^{n}$ .

Step by step solution

01

Consider the set

For the orthogonal complement $V^{鈯�}$ of the vector $V$ to be subspace of ${鈩�}^{n}$

The conditions to be followed are:

Contains a zero vector.
To be closed under addition.
To be closed under the scalar multiplication.

Consider a subspace $V$ of ${鈩�}^{n}$ .

02

Check for the first condition

Consider, $\overset{鈫�}{w} 鈭� V^{鈯�}$

Then, $\overset{鈫�}{0} 脳 \overset{鈫�}{w} = \overset{鈫�}{0}$

Thus, the orthogonal complement $V^{鈯�}$ of the vector $V$ is the subspace of ${鈩�}^{n}$ .

03

Check for the second condition

Let, $\overset{鈫�}{w} = {\overset{鈫�}{w}}_{1} + {\overset{鈫�}{w}}_{2}$

If, role="math" localid="1664201916280" $\overset{鈫�}{v} 路 \overset{鈫�}{w} = \overset{鈫�}{v} 路 ({\overset{鈫�}{w}}_{1} + {\overset{鈫�}{w}}_{2})$

Then, $\overset{鈫�}{v 路} \overset{鈫�}{w} = \overset{鈫�}{v} 路 {\overset{鈫�}{w}}_{1} + \overset{鈫�}{v} 路 {\overset{鈫�}{w}}_{2}$

04

Check for the third condition

Let, $\overset{鈫�}{w} = t 路 \overset{鈫�}{w}$ .

If, $\overset{鈫�}{v} 路 \overset{鈫�}{w} = \overset{鈫�}{v} 路 (t 路 \overset{鈫�}{w})$

Then, $\overset{鈫�}{v} 路 \overset{鈫�}{w} = \overset{鈫�}{v} 路 t 路 \overset{鈫�}{w}$

05

Final answer

$V$ Since all the conditions are satisfied, the orthogonal complement, $V^{鈯�}$ of is a subspace of ${鈩�}^{n}$ .

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Most popular questions from this chapter

Consider the vectors $\overset{鈫�}{u}, \overset{鈫�}{v}$ and $\overset{鈫�}{w}$ sketched in the accompanying figure. Find the coordinate vector of $\overset{鈫�}{w}$ with respect to the basis $\overset{鈫�}{u}, \overset{鈫�}{v}$ .

In Exercises 25through 30 , find the matrix Bof the linear transformation with respect to the basis $J = ({\overset{鈫�}{v}}_{1}, . . . . . . ., {\overset{鈫�}{v}}_{m})$ .

Consider a non-zero vector $\overset{鈫�}{蠀}$ in $\overset{}{{鈩�}^{n}}$ . What is the dimension of the space of all vectors in ${鈩�}^{n}$ that are perpendicular to $\overset{鈫�}{蠀}$ ?

In Problem 46 through 55, Find all the cubics through the given points. You may use the results from Exercises 44 and 45 throughout. If there is a unique cubic, make a rough sketch of it. If there are infinitely many cubics, sketch two of them.

53. $(0, 0), (1, 0), (2, 0), (0, 1), (1, 1), (2, 1), (0, 2), (1, 2), (3, 2)$ .

Give an example of a $4 脳 5$ matrix A with $d i m (k e r (A)) = 3$ $d i m (k e r (A)) = 3$ .

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