Eigenvectors and eigenspaces for a 3x3 matrix | Linear Algebra | Khan Academy

Eigenvectors and eigenspaces for a 3x3 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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[gembarrogo]

For those who had to google what null space is (like me), here's a quick refresher:

It is defined as the set of all vectors x that satisfy the equation Ax = 0, where A is a given matrix.

Here are some key points about the null space:

- The null space contains all solutions to the homogeneous system of linear equations represented by Ax = 0.
- It forms a vector space, meaning it is closed under both addition and scalar multiplication.
- The null space of a matrix A is a subspace of R^n, where n is the number of columns in A.
- If the only solution to Ax = 0 is x = 0, the null space consists of the zero vector alone. This subspace, {0}, is called the trivial subspace.
- The null space can provide insights into the properties of the matrix and the system of equations it represents.

[riaankorsten]

10:32 "Free real estate"

Awesome video btw!!

[FloriUchiha]

No professor of my university was able to explain properly how to determine the eigenvectors. They were just computing the end result and never explained how they came up with this result. Thank you very much, you are a genius.

[ambarishkapil]

Great explanation. Now that I have got the theory down, I will somehow need to figure out how to translate all that into Python code 😄.

[thembalethuthesacred]

wow thanks im from university of cape town, i had a problem in reducing ..now im mastering this! you're the real hero!

[RickyShehotts]

It could be either (A - lambda*I)v=0 or (lambda*I - A)v=0 . The two are the same, just differing by a multiple of (-1). Because (-1) is a constant, it can multiply into the parentheses and flip the expression inside, leaving the equation unchanged.

[finnvankolmeschate]

The real estate part really helped me out!

[pumapuma]

Thank you, my lecturer sucks. You made something he made complicated easy again.

[jkjonk]

Could you possibly do a video of why I am hearing this terminology in my Differential Equations Class?

[dieguinf]

Thanks to you I'm going to be able to pass my class.... Thank you much ;)

[VSci_]

Should have put emphasis on v3 being the free variable (row not containing a leading 1) which is why you chose v3=t. other than that very clear explanation!

[NICKNEWCOMER-nr]

Thank you so much!!! This helped me on a problem I was stuck on forever!

[Woddknife]

Thank you - You pushed my Math AND English skill through the roof - Funny that the German word: Eigenvector became a "special" word (It could have been just be translated to "own - vector") =)

[klbrumann]

Oh wow, was stressing about the last step in finding the Eigenvalues but this made it incredibly clear, thanks a lot :)

[kevinscott]

Sal I'd really enjoy it if the example you made wasn't of nullity 2, as a full matrix probably would've helped me more.

[therealalphageek]

YOU ARE THE BEST!!! :D You just cleared all the questions I sent to my professor 3 hours ago in 30 minutes ahah!!!.. YOU ARE THE BEST :D

[benswimmin]

Thank you! Very clear and comprehensible.

[autogordel]

Excellent, bad explanation at college, thank you so much for your video!

[Benjamin_Bratten]

if i had a nickel for every lab this guys helped me with id have 2 nickels

[vorapsak]

Because elementary row operations change the value of the determinant, so you'd have to "undo" them again anyway; might as well only do them once.