learned linear algebra.

planes, because they're not parallel, they're not special.

And that's what's neat about algebra.

There's a lot of names and labels in linear algebra.

linear algebra, and I'm Gilbert Strang.

And we'll do more of that in linear algebra.

Well actually, not algebra, some linear algebra.

In linear algebra, you can apply it to anything.

But in linear algebra, we like to be general.

The fundamental problem of linear algebra, which is to solve a system

And the word image is used more in a linear algebra context.

In linear algebra, linear transformations can be represented by matrices.

I actually prove it in the linear algebra playlist.

But in linear algebra, this is one of the few subjects where

If you ever get a chance to study Linear Algebra, do it, it's pretty awesome.

I'm not looking at the comment right now, but if I remember correctly, they said their linear algebra teacher said that linear algebra is the art of choosing the right basis.

But this is, I think, a really good toe dipping for linear algebra as well.

There's even a branch of mathematics that uses matrices constantly, called Linear Algebra.

It only makes sense that we have something called a linear transformation because we're studying linear algebra.

The text for the course is this book, Introduction to Linear Algebra.

Would any of you complain if this was the case about doing linear algebra?

The concept of linear combinations is central to linear algebra and related fields of mathematics.

Remember what rn was. rn we defined way back at the beginning of the linear algebra playlist.

In this first, what I'll call this is my first real linear algebra video, I want to introduce you to some concepts that you're going to see over and over in linear algebra, and if they're not introduced to you properly in the beginning, they can be very confusing.

If anything, in linear algebra, the field, or the gods of linear algebra, are kind of experts in to some degree stating simple and obvious things in convoluted and Byzantine ways.

They're actually at least a useful notation to use as we progress through our explorations of linear algebra.

History The study of linear algebra first emerged from the study of determinants, which were used to solve systems of linear equations.

linear algebra is someone will write they won't just write the set of all real numbers, they'll write Rn.

linear algebra books I looked at, they didn't specify this but you can kind of view this as the range of T.

But using a fairly straightforward, really algebraic technique, we were able to get a fairly fancy solution to this second order linear nonhomogeneous differential equation with constant coefficients.

Now I will show you another application of matrices that actually you're more likely to see in your linear algebra class when you take it in college.

Uhm, if you take linear algebra and calculus your professor might use those terms and it's good to be familiar with it.

And actually, I haven't formally defined coordinates for you yet, but I think you understand that just from your basic algebra training.

And in the terminology that you don't normally see in linear algebra a lot, you can also kind of consider it its range. The range of T.

And my sense of why, in the linear algebra world, they use this, is because you kind of imagine that this vector is being changed into that vector.

We proceed to calculate the eigenvalues of the transfer matrix formula_14,leading to the characteristic equation formula_15,where formula_16is the trace of the RTM, and formula_17is the determinant of the RTM.

The neat thing about linear algebra in general is some very seemingly simple concepts can be interpreted in a bunch of different ways, and can be shown to represent different ideas or different problems.

Well, you could go say this in a very convoluted way, but if I were to just, before you learned any linear algebra, if I said look if I have the equation f of x is equal to f of a.

And the definition of our linear transformation, or the composition of our linear transformation.

In linear algebra, an alternant matrix is a matrix with a particular structure, in which successive columns have a particular function applied to their entries.

But I'm going to be very formal and very abstract and very broad with the vector right now because the beauty of linear algebra is that it doesn't just apply to things

And for most of what we talk about in linear algebra, we're going to represent them as essentially these columns, and in the very near future, we'll represent them as rows as well.

There are more formal ways to talk about this, and there's a much more rigorous discussion of this later on, especially in the linear algebra playlist, but this is all the different values I can take on.

Independence.

linear regression.