Orthogonal Vectors and Subspaces

MIT 18.06SC Linear Algebra, Fall 2011
Instructor: David Shirokoff

A teaching assistant works through a problem on orthogonal vectors and subspaces.

License: Creative Commons BY-NC-SA

[phamhongvinh]

I find it's very helpful to have a exercise video after each lecture so you actually know how to solve the problem.
Thank OCW.

[sheikhshafayat]

I feel grateful that the internet exists. I would never have access to these fantastic lectures. Thank you, MIT!

[rickshawty]

Thanks for the help. I personally think you are the best instructor at MIT.

[AnupKumar-wked]

All of you guys are awesome. I have become fan of this linear algebra tutorial series.

[thebreath]

God you save my life. I couldn’t go to class in my college because covid and they pass all these in linear algebra and now I understand. Thanks MIT and the profesor

[supersnowva]

Excellent tutorial! Thank you David and MIT!

[qbtc]

In the 4th and 5th editions of his Intro to Linear Algebra book, Professor Strang includes worked out examples like in these recitation videos in each chapter. It's nice to have a video record nonetheless.

[qbtc]

Incidentally, he could've proceeded further with the row reduction to the full RREF to get a matrix of the form [I F] in which I is the identity and F are the free variables. Then, -F would be part of your nullspace solutions directly, ie, your bases for the orthogonal Subspace.

[quirkyquester]

very clear, very concise. Thank you David! Thank you MIT!

[andersony]

So why not just use the method Professor Strang taught in class, that to suppose free variable [x3 x4]=I? The answer comes out immediately and you never need to calculate what -2(-a+b)-2a-3b, which is slow, complex and easy to make mistakes.

[benjtheo]

Would it be the same to say that the matrix formed by combining S and S compliment you would have a matrix of rank 4 and therefore can always rewrite a vector of R4 within that space?

[bitstsunami]

Use the beautiful fact discussed in lecture 14 to solve part 1 "null space and row space are orthogonal complements in R^4 " so null space contains all vectors perpendicular to row space so simply the perpendicular subspcae to A is nullspace of A now just give basis for nullspace of A

[waichingleung]

anyone thought that he has a nice voice? lol

[wuzhenick]

Wow this is the math knowledge base for control theory to solve coupled differential equations. Am I right?

[balajikalva]

Explanation is good but he could have decomposed the matrix to RREF form from which finding null space if a simple step rather than writing the equations.

[eduardosdelarosa]

Why he didn't take those vectors into columns?

[mskiptr]

Could S^⊥ be a subspace of c_1[0 -1 1 0]+c_2[-5 1 0 1] ?
It was not specified that it has to be largest space perpendicular to S, so can we also limit ourselves? Or the notation (^⊥) automatically means 'the largest'?

[iharsh]

sure this guy, do like to flex his muscles. 😂 just kidding nyc ques.

[MrSyrian]

This subject in the book is so confusing.