7.1, the section on the first week of the class I'm concurrently enrolled in: LINEAR ALGEBRA

Greeting from chapter 7! It is entitled “Matrix Algebra” and 7.1 is entitled “Vectors and Matrices.”

Just in time, too. I have a test in my Linear Algebra class this week (because I’m doubled up in math this semester).

In this chapter, we shall be dealing with systems of linear equations. Let us consider the equations

If we isolated the coefficients on the variables we would have what is called the coefficient matrix. Just to clarify, a matrix is a rectangular array of numbers, but we’re throwing out the rectangle part and using parentheses instead.

The numbers that appear in the matrix are called entries or components.

A column vector is a matrix that contains only one column. For instance,

The x vector is called the vector of unknowns and the b vector is just the right hand side of our system. We can rewrite our system into the compact form

In C, we have two rows and three columns. Generally, if we were to have m rows and n columns we would have

The size of A is (m, n) (rows first, and then columns). A is known as an m×n matrix.

A row vector is a matrix with only one row (similar to a column vector, except row version). Also, we might need to refer to our matrices by the number of elements. For example, x would be a 3-vector and b would be a 2-vector.

Vector is used in a lot of different places and usually means something different in each topic. Because of the variety of definitions that the term “vector” can take on, we have to be careful to determine which meaning we’re using. In this chapter of the book, a vector will always be an ordered list of numbers. However, watch out for when this isn’t a thing that happens in your life.

If we look back at our matrix C, we observe it can be thought as the two row vectors (3, 2, -6) and (5, -2, 7). We can also think of C as the three column vectors

Also, as a side note: 

Wikipedia proves to us that "vector" may be the most overused word in math and science.

Also, fictional characters named Vector? 

Anyway, back to the math. 

We can add matrices if and only if they are the same size. For example,

This would be called the matrix sum A + B.

The set of column vectors with n real entries is denoted by Rⁿ. If two vectors x and y are contained in Rⁿ (which is denoted by x, y ∈ Rⁿ, where ∈ means “element of”) then the matrix sum of x and y are also contained in Rⁿ.

Something useful to note about vectors (at least for this text) is that all vectors that are parallel are equivalent.

Equivalent. (Education.com)

Another handy operation that you can do on matrices is to is multiply a matrix by a constant. For example,

If x and y are vectors in Rⁿ, then a linear combination of these vectors is a vector of the form ax + by, where a and b are real numbers.

So back to our system from the beginning of this section summary: let’s rewrite our system as a linear combination:

The right-hand side of our equation will be denoted by b. Let’s rewrite our left-hand side of our system as the product of the coefficients C and a vector of our unknowns called x:

This means we can rewrite our system entirely as Cx = b, which may not mean a whole lot to you, but it’s kind of super exciting to me since I’m in a linear algebra class.

Here’s a definition for you (but first, some exposition):

Here’s a matrix we have already seen before:

Like last time, A is a matrix of m rows and n columns. Let’s define our column vectors a₁, a₂, …, a_n, which contain all the elements in the columns. So then we can show that A has these columns by writing A = [a₁, a₂, …, a_n].

Finally, let’s define the matrix x as a vector in Rⁿ.

“We define the product Ax of the matrix A [above this definition] and the vector x [less above this definition] to be the linear combination of the column vectors of A with the coefficients from the vector x.

Ax = x₁a₁ + x₂a₂ + … + x_na_n” (277).

So we’ll have m equations and n unknowns. This system can be written as

a₁₁x₁ + a₁₂x₂ + … + a_1n­x_n = b₁

a₂₁x₁ + a₂₂x₂ + … + a_2n­x_n = b₂

…

a_m1x₁ + a_m2x₂ + … + a_mn­x_n = b_m

Our right hand side of the system can be written as

Thus our general system can be rewritten as Ax = b. If Ax is defined, then number of columns in A must equal the number of rows in x.

Something that helped me with this is something I can possibly recreate using Paint:

I hope that kind of makes it better for you to understand.

Here’s a theorem about multiplication:

“Suppose A is an m x n matrix, x and y are in Rⁿ, and a is a number. Then

1. A(ax) = aAx

2. A(x + y) = Ax + Ay” (279).

Multiplication by A is called a linear operation. Also, no proof was given in the section (yes!). It’s fairly easy to prove these things using the definitions.

Multiplication of matrices A, B, and C is…

Associative: A(BC) = (AB)C

Distributive: A(B + C) = AB + BC or (B + C)A = BA + CA

However, it is a rare and wondrous occasion indeed for matrix multiplication to be commutative. This means it probably won’t be a thing to have AB = BA. You can make two matrices that can be multiplied both forwards and backwards (I would recommend a square matrix, like a 2 x 2 or a 3 x 3). In any event, unless you choose the zero matrix (zeroes in every place) then you will find that you’re either really amazing at picking random numbers or the commutative properties don’t happen often for multiplying matrices.

Here’s the identity matrix:

As you can see, there are ones on the diagonal and zeroes every other place. Something cool to note about the identity matrix is that Ix = x for every vector x. This means that IA = A and BI = B.

One more thing to leave you with concerning this section is the transpose of a matrix.

Here’s our matrix A:

Here’s the transpose:

So you just flip the matrix along its main diagonal. Here’s an example:

Something nice about this is that a matrix with m rows and n columns will have a transpose of n rows and m columns.

Okay, that’s about it for 7.1. Just to let you know, we have 6 more sections of chapter 7. I’ll see you when I see you.