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### Course: Linear algebra>Unit 3

Lesson 1: Orthogonal complements

# Orthogonal complements

Orthogonal Complements as subspaces. Created by Sal Khan.

## Want to join the conversation?

• After , should all the rn-vectors should be rn-transpose-vectors like they were before? Or, since he "untransposes" his r-vectors at , should all the r-vectors he wrote between and be non-transpose vectors as well? I'm confused.
• The "r" vectors are the row vectors of A throughout this entire video. "x" and "v" are both column vectors in "Ax=0" throughout also.

The dot product of two vectors is not affected by the row or column context of either vector.

The "T"s he uses on vectors here are useless and confusing. @ he's not using the row vectors any differently than he did anywhere else in this video.
(1 vote)
• At in the video, isn't A really A transpose? And what Sal calls r1T, r2T aren't they really the transpose of Column 1 Column2, so maybe they should be labelled c1T, c2T? Because the r1T suggests the transpose of a row to a column.
• The notation confused me for a bit also, but i think he just wants to clarify that any row matrix could be written as a transpose of the column matrix ed v_1T which we are more used to.
But saying that it shoud be from r1T to r_nT not r_mT
• at is every member of N(A) also orthogonal to every member of the column space?
• every member of N(A) also orthogonal to every member of the column space of A transpose.
• Is it possible to illustrate this point with coordinates on graph?
• May you link these previous videos you were talking about in this video ?
• I have a question which gave me really big confusions for past few days....

1.I want to know is ortogonal complement actually a set of vectors which are ortogonal on some other set(subspace)?

2.What is difference beetween ortogonal complement and ortogonal component?If ortogonal complement is set of ortogonal vectors on some set
is component actually 1 of the vectors inside the complement?

3.Is process of getting ortogonal set(complement) of vectors on some particular set called "gran schmidt process" ?

4.And if 3rd is correct then what is NULL space?
Thanks
• The orthogonal complement is a subspace of vectors where all of the vectors in it are orthogonal to all of the vectors in a particular subspace. For instance, if you are given a plane in ℝ³, then the orthogonal complement of that plane is the line that is normal to the plane and that passes through (0,0,0).

The orthogonal component, on the other hand, is a component of a vector. Any vector in ℝ³ can be written in one unique way as a sum of one vector in the plane and and one vector in the orthogonal complement of the plane. The latter vector is the orthogonal component.

The Gram-Schmidt Process is not the process of getting the orthogonal complement of a subspace from the original subspace. It is actually a method of creating an orthonormal coordinate system. For more information about orthonomal bases, Sal explains it at https://www.khanacademy.org/math/linear-algebra/alternate_bases/orthonormal_basis/v/linear-algebra-introduction-to-orthonormal-bases.

The 3rd question is not correct, but I'll answer this question anyway. A null space is the set of vectors which when multiplied by a matrix gives the 0 vector.
• What's the "a member of" sign Sal uses at and, again, about 10 seconds later. It looks like a smaller-or-equal sign. Is this a common thing?
• This is the notation for saying that the one set is a subset of another set, different from saying a single object is a member of a set. This notation is common, yes. This is a short textbook section on definition of a set and the usual notation: http://linear.ups.edu/html/section-SET.html
• is confusing, the rest is actually okay. So...r1 through to rm are row vectors...but wait, they are transposes of row vectors....doesn't that mean they are column vectors...? But then he writes it in that way....Please somebody help to explain...why they are just the same thing as ordinary Matrix A, this been bugging me and is starting to waste my time so I will just leave the question here and if you have a good idea about how it works please tell me :) Thank you for reading my question ^_^
• At , is it supposed to be rn transpose or rm transpose multiplied by x ?
A matrix has m rows and he is essentially multiplying the rm th row by vector x.
• Try it with an arbitrary 2x3 (= mxn) matrix A and 3x1 (= nx1) column vector x. You'll see that Ax = (r1 dot x, r2 dot x) = (r1 dot x, rm dot x) (a column vector; ri = the ith row vector of A), as you suggest.
(1 vote)
• In this video, Sal examines the orthogonal complement.
Complementary angles add to ninety degrees.
Is the term complement used in other contexts to represent perpendicularity, or does it just mean something that goes with it, as in "This wine complements this food."?