Finding eigenvectors and eigenspaces example | Linear Algebra | Khan Academy
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Finding the eigenvectors and eigenspaces of a 2x2 matrix
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Linear Algebra on Khan Academy: Have you ever wondered what the difference is between speed and velocity? Ever try to visualize in four dimensions or six or seven? Linear algebra describes things in two dimensions, but many of the concepts can be extended into three, four or more. Linear algebra implies two dimensional reasoning, however, the concepts covered in linear algebra provide the basis for multi-dimensional representations of mathematical reasoning. Matrices, vectors, vector spaces, transformations, eigenvectors/values all help us to visualize and understand multi dimensional concepts. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.
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"For any" -Sal Khan
My hero, you know I skip class and watch these videos which are far more helpful! Your explanation is very easy to follow
Sal you have seriosuly a gift of teaching. I just need to sit back and watch your videos and suddently the urge to get a book and take notes hits me because, I learned something else. With you learning is so simple. Im not gonna lie I hope you win the Nobel prize in the future, you're doing a huge favor to everyone in the world
Thanks man! I take ages to understand stuff, and rewinding your explanation at the end over and over again really helped a lot! The thing with live lectures for me is that if i miss a sentence I may as well go home, because I'm confused for the rest of the lecture.
This is one of best interpretation of eigen value and eigen space.
Everyone seems they reached out to this right before a test! I'm just glad I could do my homework now :D
If you watch the proof video (which I highly recommend) you will see that at the beginning when Av=&v we can either write Av-&v=0(vector) OR &v-Av=0(vector). Proceeding with the rest of the proof should convince you that taking the determinant of either (A-&I) or (&I-A) will work.
Watch his video before lecture makes me feel smart during lesson😊
So how does this dude know everything?
reading i guess
@@Rjsipad he is a Harvard computer science and math major
Because he drinks
dude i want to thank you so much, today in math class I had to do exactly what you are teaching in this video and i got 20 out of 20 :), thus passed mathclass :P! ty man!
Thank you so much from Italy! Now maybe I could pass this linear algebra exam!
Thank You so much I looked at my professor like he was speaking another language when he tried to teach this subject and you made it sooo clear
this is the most helpful video i have ever watched...thank you! you dont know how much stress you just took off of my shoulders!
i literally applauded for this guy after this video. Lifesaver.
You can make a pigeon understand linear algebra, wonderful.
literally the same example in my practice final. Thanks for clarifying some little things. You're a beast
Simply extraordinary presentation by Mr. Khan! I think you learnt this stuff from Dr. Strang at MIT. Thank you for rendering your services for less privileged.
Thanks for posting these, I had an awesome linear algebra teacher so I never went to class, and I'm still going to make an A.
2:34 Theres a glitch in the matrix...
Yeah, sometimes there is
Aman Khosa 2:53 - 3:10
Thanks a ton. I love u dude..
I didn't attend this lecture in college and tommorow I have a class test on it. Really helpful. :)
Thank you so much, this really helped me out.
I just don't get it by looking at theorems at all, I need examples.
The geometric explanation give powers to understand everything, I feel like I can see more one dimension. Now I have Just 6 missing.
You're a life saver, Sal Khan.
2:53 - 3:10 really emphasizing the new "eignenspace" term, great! :)
Thank you so much!!!. This video is super helpful!!!
Thanks very much . After many years I think a gonna get it now about eigenvectors
Lennart
Finally I have got a good teacher 😊😊😊
very helpful, thank you! I don't understand how you can write so neatly with a mouse...props.
wow you really are amazing. all my maths and econ i learn from you...thanks a bunch
I really wish my professor made it this easy. I shouldn't be struggling this much with this material.
Who needs to go to lectures when you have these?
These comment section has people who aced this exam, graduated, did phd, now successful, married life and have kids lmao
You are a really good teacher. Thanks so much.
you are totally awesome, this video is so great! I sat for 2 hours in math lecture, no idea what the hell my prof. was talkin about. I watch your 10-15 min vids and get the thing right away... you ruLE! cheers from Austria
if you know eigen values and the eigenvectors you can multiply the modal matrix (matrix of eigenvectors) times the spectral times the modal inverse and I am pretty sure that will give you the matrix A... like so if M^-1*A*M = AS, where AS is the spectral matrix (a diagonal matrix made of the eigenvalues), then the inverse should be true, M*AS*M^-1 =A ....
@zxtreme09 I am not talking about the matrix: top (1 2), bottom (4 3) provided in the video, but the matrix that happsider asked about: (-3 -4) on the top and (1 0) on the bottom.
Tomorrow is my linear algebra final. The last math class I have to take. FUCK YEAH! WOOHOO!
Great video, You might want to mention using "parametric vector form" I noticed this because I use it frequently to determine eigenvectors and bases! Handy for diagonalization! Thanks again Khan.
I am really confused about Null Space. Sometimes it is a span of a vector, but sometimes it is only set of one vector. I don't know when to calculate what. When is the null space a span and when is it just one vector? And how do I differentiate when to calculate what?
Thanks a lot
the CORRECT equation for eigenvalues is **det(A - lamda(I)** NOT "det(lamda(I) - A)" according to Gilbert Strang.
PTranPhotography it shouldn't matter lol. You can multiply both sides by -1 because the determinant must be 0 for it to be an eigenvalue.
nice one.... thanx for this video... it really helped out.....
im officially less fucked for my linear algebra final tomorrow.. thanks!!
you are the absolute best
My final is in 3 weeks! Feels good though!
Could you do a video on generalised eigenvector?
@ NJMetsHero
That doesn't matter since you use the determinant. All the signs are opposite, but by multiplying negatives, you still get a positive.
Which is the correct one: det(yI-A) or det(A-yI) ?
Thank you
when would you ever have two different vectors spanning an eigen space? how would you get this from reduced row echelon method?
@happsider (& means lambda) If there are no mistakes in my calculations, you have got two values for eigenvalue: & = -1 and & = -3. When substituting them into [A-&I][v] = 0 it appears that the only possibility (in both cases) for v is to be zero vector (0). Since 0 cannot be an eigenvector, I conclude that there are no eigenvectors nor eigenvalues for the matrix A you provided.
If I am wrong, please, can anyone correct me?
Thank You for this vid, would be better if i watched it BEFORE i failed my finals :/ oh wells at least i have a supp exam to pass now!!
I love this man :D
So what if the matrix to find the eigenvectors has two different results?? I had a matrix (-3 -4) on the top and (1 0) on the bottom. The top row and the bottom row yield different values for v1 and v2 so which one do i choose?!
Great video! Thanks for making it!
is there a difference between V1= -V2 and V2= -V1? The Eigenvectors would be [1,-1] or [-1,1]. I'm guessing there's no difference as the both lie on the same vector?
Great teacher!
These videos basically saved my GPA.
I wish I had watched this before my quiz...
in a 2x2 matrix, what is after you do (lambda I - A) = 0 you get a matrix thats with first row [ 2 0] and second row [2 0] how do you find eigenvector in that case?
How do you find the multiplicity 2 of eigenvalue?
a question: is the echelon form necessary, cause i put the Lambda = 5 respectively -1 in the A*v=lambda*v and i get the same results for this exemple but i have 2 equations instead of 1... is the raw echelon form necessary for other matrices?? perhaps its a dumb question but maths confuses me.
whats up with the reduced row echelon form you do starting at 5:21
doesnt the result matrix have to be a identity matrix with a diagonal out of 1's ??
Put that matrix into reduced row echelon form, then solve for the nullspace. C:
Dr. Khan there is a glitch in the matrix row 1 column 1 @5:41 sign error
If a transformation matrix with respect to a basis other than standard basis is given do I need to find the matrix with respect to standard basis and find the eigen values ??
in the book det[A-&I]=0 what is the difference?
at 11:00.. if i choose to let my v1=t and v2=-t .
will my eigenvector t[1 -1] and yours t[-1 1] both be correct ??
you can teach better than my teacher...
I don't get what is meant by transformation at the end of the video... can someone explain please? what is the eigenvector transformed into and why is it transformed?
It works either way I think
does it matter if it is (A- lamda Ide. vector)
Can i Take:
N(I - rref(A))
?
hi there, great vid, so helpful. I was wondering, I am stuck on an unusual question involving eigenvectors. It asks to 'Show that if a matrix M has an eigenvector x corresponding to the eigenvalue lambda, x is also an eigenvector of the matrix M^2, and find the corresponding eigenvalue.What is the analogous result for M^n, where n is any positive integer?
I would be very grateful if someone could help me out. I'll keep trying though. :)
prefection
You can do both.
@TooNarrowToContain thank you very much, this helped a lot :)
@TimeIsTheMatter you came all the way to Salman Khan's video on Eigenvectors to comment on a year old post? Please do yourself a favor, and take a seat, and learn your Eigenvectors before the final.
because he wrote it as t[1 -1], by setting t = -1 you get [-1 1] as your result. My wager would be yes.
Thank you so much. Finally I can burn my book from school because the explanation is so bad. You made it clear for everyone who's not that good in this stuff.
mine is tomorrow! xD or well later today...
ikedamiler in 15 minutes I’ll be in your situation
say negative instead of minus
lol. after watching this i can say. my professor has no idea in Linear Algebra :D
you forgot the negative sign in front of the 4
much better than strang no offense prof
@TimeIsTheMatter cool
if you watch how he manipulated the Ax=Lx you'd realise they're the same thing
I UNDERSTAND EVERYTHING xD
no i dont but hopefully with his videos I will soon =]
well Khan sir how didi u get lamda =5
(lamda-1)(lamda-3)-8
for any... for any.... for any...
why the -4 turns to 0
after 14 years
for any, for any, for any, for any....
Final exam in one hour.
all dem weedz bro.
@TimeIsTheMatter My god chill out...
If you can ...... please slow down while teaching so our thoughts process can catch up .
This is a mathematics not English
LEGEEEENDDDDD
#thepokigur i'd marry you if you want :D
Marry me!!!