(a) find the orthogonal complement S⊥, and (b) find the direct sum S⊕S⊥.S is the subspace of R5 consisting of all vectors whosethird and fourth components are zero.

Answered: (a) find the orthogonal complement S⊥,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

(a) find the orthogonal complement S^⊥, and
(b) find the direct sum S⊕S^⊥.

S is the subspace of R⁵ consisting of all vectors whose
third and fourth components are zero.

Expert Solution

Step 1

It is given that S is the subspace of $ℝ^{5}$ consisting of all vectors whose third and fourth components

are zero.

Thus, we have $S = \{(x, y, z, 0, 0) x, y, z \in ℝ\}$ . We have to find the orthogonal complement of the

subspace S.

Step 2

Part a

We know that $S^{⊥} = \{v = (a, b, c, d, e) <v, w> = 0 where w = (x, y, z, 0, 0)\}$ .

Now,

$\begin{array}{rcl} w & = & (x, y, z, 0, 0) \\ w & = & x (1, 0, 0, 0, 0) + y (0, 1, 0, 0, 0) + z (0, 0, 1, 0, 0) \\ ∴ S & = & span \{(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0)\} \end{array}$

So we get,

$\begin{array}{rcl} <v, w> & = & 0 \\ (\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \end{matrix}) (\begin{matrix} a \\ b \\ c \\ d \\ e \end{matrix}) & = & (\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}) \\ (\begin{matrix} 1 \cdot a + 0 \cdot b + 0 \cdot c + 0 \cdot d + 0 \cdot e \\ 0 \cdot a + 1 \cdot b + 0 \cdot c + 0 \cdot d + 0 \cdot e \\ 0 \cdot a + 0 \cdot b + 1 \cdot c + 0 \cdot d + 0 \cdot e \end{matrix}) & = & (\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}) \\ (\begin{matrix} a \\ b \\ c \end{matrix}) & = & (\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}) \\ a & = & 0, b = 0, c = 0 \end{array}$

Thus, $S^{⊥} = \{(0, 0, 0, d, e) d, e \in ℝ\}$ .

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