Get the answer to "Let U=Sp{(3,3,1)} How can I find the Orthogonal..."

College
Algebra
Linear algebra
Vectors and spaces

garnentas3m

Answered question

2021-12-11

Let

U = S p {(3, 3, 1)}

How can I find the Orthogonal complement ? Im

Answer & Explanation

Kirsten Davis

Beginner2021-12-12Added 27 answers

The plane of equation $P \equiv 3 x + 3 y + 1 z = 0$
It has dimension equal to 2 and every vector $u = (x, y, z) \in P$ is such that $< u, v \geq 0$ where $v = (3, 3, 1)$

chumants6g

Beginner2021-12-13Added 33 answers

Since U has only one dimension, it is indeed true that A will have only one line. Hence, the orthogonal complement $U^{⊥}$ is the set of vectors $x = (x_{1}, x_{2}, x_{3})$ such that.
$3 x_{1} + 3 x_{2} + x_{3} = 0$
Setting respectively $x_{3} = 0$ and $x_{1} = 0$ , you can find 2 independent vectors in $U^{⊥}$ , for example $(1, - 1, 0)$ and $(0, 1, 3)$ . These generate $U^{⊥}$ since it is two dimensional (being the orthogonal complement of a one dimensional subspace in three dimensions). Hence, we can conclude that
$U^{⊥} = Span {(1, - 1, 0), (0, 1, - 1, 3)}$
Note that there would be many (infinitely many) other ways to describe $U^{⊥}$

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