Starting things off with creating a foundation with matrices. This lecture covers what a matrix is, types of matrices, and matrix algebra. The video also includes examples (which is why it's a long one lol).
In this lecture, you will see that you can use matrices to solve systems of linear equations using Gaussian elimination as well as learning about the kernel of a matrix.
This examlpe will solve a system of linear equations that contains a unique solution (one solution).
This second example will be solving a system of linear equations that containes infinitely many solutions which will introduce the kernel.
This final example of for systems of linear equations looks at a system that has no solutions.
This lecture will introduce the inverse matrix and the steps of how to find them.
A linear short? What is that??? Well it's a small series where I explain somethings that can't fit in a video without taking a lot of time. Anyways, this short video will discuss about the rank of a matrix as well as pivots.
This example will show you a simple method to finding inverse of a 2-by-2 matrix as well as the general way of finding any inverse matrix.
Linear proof?!?! Ayo, what is this!?!? Well they are proofs to help us get a better understanding on why somethings just work. This proof will derive the formula of finding the inverse of any 2-by-2 matrix.
As we wrap up the chapter, this final lecture goes more in depth with determinants, how to solve them, and the unique properties that they have.
This video will go through the process of manipulating a matrix with a given determinant to find the determinant of another matrix. This video will also cover 1 method of findng the determinant of a given matrix.
Finding the determinant for a 2-by-2 matrix is easy, but what about higher dimensional matrices? This video will calculate the determinant of 3-by-3 matrices using square reduction (which is the general method) and through expansion which is a method exclusively for 3-by-3 matrices.
This video will close the chapter on matrices with one more method in solving for determinants and a brief review in finding the determinants.
This new chapter will be looking at vector spaces and subspaces. This will video will define what a vector space and subspace are.
This video will look at a set of examples and check to see if each given set is a subspace.
This lecture will be introducing linear independence and learn what makes a vector linearly independent as well as discuss about spans and basis.
This video will explain how to find a basis for a given matrix subset.
A challenging example that proves the fact that there is a subset of a given vector space and as well as finding its basis along with the dimension of the basis.
This video will show how to check if a given vector spans and also a way to write a polynomial in vector/matrix form.
There are many ways to ask if a set spans and finding basis... luckily, this video contains more ways these questions can be asked :D
Shifting the tide with all the spans and basis, this example covers if a vector is a linear combination of a given set.
The last example from this lecture!!! How do you make a set have vectors that are linearly independent from each other? Well for starters, you need a free variable (we will call k).
Did you know a subset can have multiple set of basis?? Well my friend, this lecture will introduce the concept of changing from one base to another, why they are important, and as well as the rank-nullity theorem along with the consequences this theorem brings. Brace yourself because this is going to be a lot of information to take in!
This short video will give a visual representation on changing basis with the standard basis.
This heavily detailed example will look at how you convert any basis into standard basis form as well as visually showing how we can get to the answer
This example looks at 2 methods in finding the transition matrix, or how I like to call it, our "translator" (spoiler alert: one is better than the other).
Continuing on with calculating the transition matrix, I will point out a neat little observation as well as solve an example with two "wacky" basis.
Slowing closing the chapter as this video shows, how to calculate the row and column space, as well as finding the rank and nullity of a given matrix.
As the chapter closes for vector spaces and subspaces, this last example will show how the rank-nullity theorem can help determine if a system of linear equations has a solution.
This chapter will focus all about linear transformations. What they do and some interesting properties.
Linear transformations are low-key hard lol, so I will be taking it slow and do some examples on images and pre-images then showing a simple linear transfomration example.
That last example video wasn't too bad wasn't it?? Well lets continue building up that momentum and find the image and/or pre-image of a given linear transformation.
Switching gears with the images, this video will show the process of checking to see if a function is a linear transformation... why does this process look so familiar?!?
Part 2?!?! Ayo?!?!? Yes there is a part 2 which looks at more examples that you didn't even know they could ask ;-;
We are finding the image again?! Have we not done that already??? We have... but have you tried finding the image without being given the function? Crazy how its doable, it's almost like you are a becoming a detective and working to solve this case.
Okay detective, we have another case that's much more difficult. Let's crack this case now!
We are approaching the last couple chapters on linear algebra (YAAAAYY). Anyways, we are going to take a step back and go do some easy stuff, some of which you have probably done before. This video will discuss about vectors and their lengths, and dot product.
This simple video will focus on finding the magnitude and unit vector of different dimensional vectors.
Wanna know where the dot product came from?
Continuing with inner product spaces, these examples will look at dot products and distance between vectors.
After a review, for some of you, on dot products and lengths, we will be expanding the concept of the dot product to what is known as the inner product space, and generalize other concepts that we once knew.
Before calculating inner products of different vectors, we are going to prove that a given function is an inner product. You do need a bit of a calculus background, and if you don't have one, that's okay since it won't be calculus heavy.
A bit more calculus heavy, but still goes deep into the inner product of two functions. Functions?!?! Yup, you can do that.
Okay enough with the calculus, lets do some more inner product examples :D
Our last lecture for the chapter covers the concept of orthonormal bases and an algorithm that helps us create an orthonormal basis set for any dimension.
These 2 examples will showing you how to check if a set is orthogonal or orthonormal and learn how to find a coordeinate vector relative to an orthonormal base.
This video will solely focus on the Gram-Schmidt process and will also contain the alternative method.
This last example will continue with the Gram-Schmidt process as we close the inner product spaces chapter.
In this last chapter, we will be discussing about the eigenvector and eigenvalue, how to find them, eigenspaces, and the eigenvalue problem.
This example will show you how to verify that the eigenbros belong to the matrix.
This video will go in detail in finding the eigenbros given a matrix.
More examples with the eigenbros :D
This video will focus on a much quicker method in finding the eigenvector... the plug-and-chug method :D
The final example looks at a 3-by-3 matrix and solves the eigenvalues using the characteristic equation (woah we about to solve a cubic polynomial) and finding the corresponding eigenvectors.
We are getting close to finishing this chapter!! This lecture will talk about the process of diagonalizing a matrix which can be very useful as you will see in the upcoming examples.
As always, let's slowly ease into this concept and first check to see if a matrix is diagonalizable given a matrix and an invertible matrix.
Now that we know how to verify if a matrix is diagonalizable, we will now begin the process of diagonalizing a matrix. This means we also have to find the invertible matrix S and then verify it :D
We have diagonalized a 2-by-2 matrix. That was soooo easy. now lets do a 3-by-3, so take a trip into my head to see my detailed thought process of how I solve this. *Side note: it doesn't actually take this long to solve this problem in reality so don't freak out with the length of the video.
We have been diagonalizing matrices for quite a while, but did you know you can diagonalize linear transformations?!?!
This video will show the cool things you can do with diagonalizations :D
The second part of matrix powers as this one deals with a 3-by-3 matrix :D
Ever tried to take e and raise it to some matrix??
We are almost done with linear algebra!!!! This lecture will get everything we have been learning theses past couple lectures to show the cool things you can do with them to make your life much easier.
Another linear proof that looks into the real spectrum theorem. This is not the general proof since it is beyond linear algebra, but I will show that a symmetric 2-by-2 matrix is always diagonalizable.
This video will show why an orthogonal matrix contains a base of orthonormal column vectors.
We are back with actual examples now :D As usual, the first example will be simple and will look at the eigenvalues and eigenspace of symmetrical matrices to show that they all give real eigenvalues.
This video will focus on checking if a matrix is an orthogonal matrix.
For our last example in this lecture, we will be diagonalizing symmetrical matrices with an orthogonal matrix.
OUR LAST LECTURE FOR LINEAR ALGEBRA!!!!!!! This lecture talk about different types of eigenvalues and as well as the Jordan-Canonical Form.
This video will calculate eigenvectors with complex eigenvalues just to show that it is no different than if they were real. It can also take less time :D
Did you know that you can "predict" how the Jordan normal form will look simply by using the eigenvalues you found on a given matrix ;-;
OUR LAST EXAMPLES FOR LINEAR ALGEBRA!!! This last example video will show how to find our invertible matrix so we can have an almost diagonal matrix, known as the Jordan-Canonical Form. A farewell to you guys for now ;)