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Let's get started with our linear algebra review.

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In this video I want to

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tell you what are matrices and what are vectors.

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A matrix is a

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rectangular array of numbers

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written between square brackets.

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So, for example, here is a

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matrix on the right, a left square bracket.

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And then, write in a bunch of numbers.

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These could be features from

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a learning problem or it could

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be data from somewhere else, but

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the specific values don't matter,

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and then I'm going to close it with another right bracket on the right.

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And so that's one matrix.

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And, here's another example of

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the matrix, let's write 3, 4, 5,6.

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So matrix is just another

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way for saying, is a

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2D or a two dimensional array.

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And the other piece

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of knowledge that we need is

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that the dimension of the

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matrix is going to

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be written as the

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number of row times the number of columns in the matrix.

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So, concretely, this example

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on the left, this

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has 1, 2, 3, 4

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rows and has 2 columns,

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and so this example on the

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left is a 4 by

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2 matrix -  number of rows by number of columns.

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So, four rows, two columns.

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This one on the right, this matrix has two rows.

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That's the first row, that's

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the second row, and it has three columns.

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That's the first column, that's the

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second column, that's the third

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column So, this second

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matrix we say it is

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a 2 by 3 matrix.

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So we say that the dimension of this matrix is 2 by 3.

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Sometimes you also see this

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written out, in the case

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of left, you will see this

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written out as R4 by 2

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or concretely what people

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will sometimes say this matrix

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is an element of the set R 4 by 2.

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So, this thing here, this

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just means the set of all

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matrices that of dimension

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4 by 2 and this thing

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on the right, sometimes this is

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written out as a matrix that is an R 2 by 3.

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So if you ever see, 2 by 3.

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So if you ever see

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something like this are 4 by

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2 or are 2 by 3,

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people are just referring to

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matrices of a specific dimension.

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Next, let's talk about how

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to refer to specific elements of the matrix.

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And by matrix elements, other than

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the matrix I just mean

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the entries, so the numbers inside the matrix.

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So, in the standard notation,

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if A is this

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matrix here, then A sub-strip

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IJ is going to refer

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to the i, j entry,

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meaning the entry in

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the matrix in the ith row and jth column.

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So for example a1-1 is

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going to refer to the entry

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in the 1st row and

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the 1st column, so that's the

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first row and the first

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column and so a1-1

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is going to be equal to

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1, 4, 0, 2.

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Another example, 8 1

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2 is going to refer to

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the entry in the first

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row and the second

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column and so A

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1 2 is going to be equal to one nine one.

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This come from a quick examples.

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Let's see, A, oh let's

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say A 3 2, is going to refer

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to the entry in the 3rd

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row, and second column,

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right, because that's 3 2

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so that's equal to 1 4 3 7.

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And finally, 8 4 1

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is going to refer to

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this one right, fourth row,

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first column is equal to

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1 4 7 and if,

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hopefully you won't, but if

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you were to write and say

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well this A 4

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3, well, that refers to

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the fourth row, and the

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third column that, you know,

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this matrix has no third

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column so this is undefined,

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you know, or you can think of this as an error.

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There's no such element as

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8 4 3, so, you know, you

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shouldn't be referring to 8 4 3.

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So, the matrix

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gets you a way of letting

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you quickly organize, index and access lots of data.

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In case I seem to be

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tossing up a lot of

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concepts, a lot of new notations

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very rapidly, you don't need

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to memorize all of this, but

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on the course website where we

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have posted the lecture notes,

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we also have all of these definitions written down.

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So you can always refer back,

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you know, either to these slides,

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possible coursework, so audible lecture

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notes if you forget well, A41 was that?

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Which row, which column was that?

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Don't worry about memorizing everything now.

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You can always refer back to

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the written materials on the course website, and use that as a reference.

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So that's what a matrix is.

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Next, let's talk about what is a vector.

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A vector turns out to be a special case of a matrix.

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A vector is a matrix

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that has only 1 column so

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you have an N x 1

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matrix, then that's a remember, right?

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N is the number of

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rows, and 1 here

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is the number of columns, so, so

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matrix with just one column

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is what we call a vector.

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So here's an example

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of a vector, with I

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guess I have N equals four elements here.

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so we also call this

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thing, another term for

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this is a four dmensional

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vector, just means that

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this is a vector with four

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elements, with four numbers in it.

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And, just as earlier

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for matrices you saw this

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notation R3 by 2

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to refer to 2 by

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3 matrices, for this vector

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we are going to refer to this

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as a vector in the set R4.

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So this R4 means a

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set of four-dimensional vectors.

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Next let's talk about how to refer to the elements of the vector.

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We are going to use the notation

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yi to refer to

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the ith element of the

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vector y. So if y

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is this vector, y subscript i is the ith element.

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So y1 is the

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first element,four sixty, y2

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is equal to the second element,

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two thirty two -there's the first.

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There's the second.

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Y3 is equal to

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315 and so on, and

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only y1 through y4 are

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defined consistency 4-dimensional vector.

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Also it turns out that

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there are actually 2 conventions

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for how to index into a

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vector and here they are.

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Sometimes, people will use

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one index and sometimes zero index factors.

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So this example on the left

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is a one in that

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specter where the element

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we write is y1, y2, y3, y4.

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And this example in the right

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is an example of a zero index

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factor where we start

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the indexing of the elements from zero.

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So the elements go from a zero up to y three.

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And this is a bit like the

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arrays of some primary languages

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where the arrays can either

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be indexed starting from one.

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The first element of an

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array is sometimes a Y1,

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this is sequence notation I guess,

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and sometimes it's zero index

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depending on what programming language you use.

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So it turns out that in

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most of math, the one

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index version is more

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common For a lot

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of machine learning applications, zero index

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vectors gives us a more convenient notation.

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So what you should usually

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do is, unless otherwised specified,

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you should assume we are using one index vectors.

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In fact, throughout the rest

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of these videos on linear algebra

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review, I will be using one index vectors.

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But just be aware that

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when we are talking about machine learning

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applications, sometimes I will

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explicitly say when we

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need to switch to, when we

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need to use the zero index

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vectors as well.

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Finally, by convention, usually

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when writing matrices and vectors,

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most people will use upper

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case to refer to matrices.

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So we're going to use

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capital letters like

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A, B, C, you know,

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X, to refer to matrices,

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and usually we'll use lowercase,

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like a, b, x, y,

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to refer to either numbers,

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or just raw numbers or scalars or to vectors.

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This isn't always true but

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this is the more common

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notation where we use

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lower case "Y" for referring

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to vector and we usually

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use upper case to refer to a matrix.

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So, you now know what are matrices and vectors.

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Next, we'll talk about some

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of the things you can do with them