Welcome to Module 5. Today, we'll be looking at two really important concepts in linear algebra that you will undoubtedly need to know if you want to pretty much do anything in linear algebra, talk about linear algebra, or know the basic terminology of linear algebra going forward. First off is A transpose, and second will be the inner product or dot product, depending on who you ask, they'll either like to call it dot product or they'll like to call it an inner product. First, for A transpose, essentially what A transpose is, and this will work with vectors because vectors can be thought of as just a matrix with either one row or one column. But in general, it's a matrix operation in which the rows and the columns are flipped. For example, if I have A, let's call this matrix A, and it's 3, 2, 1, 4, 5, 6. Then from here, A transpose, usually denoted with a superscript of T. Notice here that this is three rows and two, this is a three by two matrix. A transpose, since we're flipping it, we flip it about the diagonal essentially is what we're doing here, but if you want to think about it, like take the columns and make them rows, take the first column, make it the first row. We can think about it like that and we'll end up with a two by three matrix. The first column is 3, 2, 1, so now my first row is 3, 2, 1. My second column is 4, 5, 6, so now my second row is 4, 5, 6. It's as simple as that. If we had a vector, let's say we had a vector 1, 2, 3, and we want to call it C, it doesn't matter what we call it, then C transpose will just be equal to a row vector of 1, 2, 3. If our original vector was a row vector and we take the transpose, we would get a column vector. Now, the dot product or inner product is of two different vectors. Now, we've talked about A transpose, let's move a little bit toward the dot product here. We already have this U and this V vector, these are column vectors. What the dot product is, is let's say U dot V, and that's usually why it's casually known as the dot product. What that is, is I want to multiply U and V together, but they're both column vectors. What you do is, in order to get this to work, this is a three by one, I will need to do something to it in order to make this work. I would take U transpose, what that would do is, this would turn a three by one into a one by three. Now I have a one by three here, U transpose as a unit, is a one by three now because I've taken the transpose of this U vector and I want to multiply it by V, which is a three by one, I'm not touching V. The inners match, we can multiply. Good. That's a little check-mark, we can multiply. The remaining thing that we're going to get is a one by one, pretty much a number. A matrix that's one by one is just a number. When we do this, remember this is the original function right here, U dot V or U inner product with V. Take the first vector transpose and then multiply it by V in the traditional sense of multiplication. We would have 2, negative 5, negative 1, that's U transpose. V is 3, 2, negative 3. When we multiply that out is 2 times 3. Remember, this is a one by one. I'm just going to do the algebra in here, but this is all one thing. It's 2 times 3 plus negative 5 times negative 2, negative 5 times 2 plus negative 1 times negative 3, that's 6, this is negative 10, which will let me a negative 4 plus 3, I get negative 1. In general, when we deal with dot products, we'll end up with a one by one matrix, but really we're just looking for the number that comes out of there. In this case, it's a negative 1. Now, you can think about it. If you think about what's happening here, we're multiplying the first entry times the first entry, second entry times second entry, and third entry by third entry here. Really, if we take U and we transpose it and multiply it by V, or if we take V and transpose it and we multiply it by U we'll end up getting the same exact entries being multiplied by each other and then added. We'll find that V transpose U is equal to U transpose V. Essentially what that's saying is, which is an important thing to realize here, which is U dot V equals V dot U. That's it. Hopefully, you're comfortable with transpose. Transpose is a pretty simple concept, but as we go forward in Module 5, we'll definitely need to know it and we'll definitely need to be able to implement it with matrices and vectors. Dot product is an extremely useful idea in linear algebra, and to leave any class in linear algebra without knowing what a dot product is would be a crime. We've learned both of these and next time let's go a little bit more in-depth with them.