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Master Solving Systems of Linear Equations by Graphing

Linear Algebra: Master Core Concepts & Applications

Linear Equations with Matrices

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Row reduction and echelon forms

Linear combination and vector equations in \(R^n\)

Matrix equation Ax=b

Solution sets of linear systems

Application of linear systems

Solving systems of linear equations by graphing

Notation of matrices

The three types of matrix row operations

Representing linear system as a matrix

Representing a linear system as a matrix

Solving a linear system with matrices using Gaussian elimination

Linear Transformation

Linear independence

Image and range of linear transformations

Properties of linear transformation

The matrix of a linear transformation

Matrix Operations

Adding and subtracting matrices

Multiplying a matrix by a scalar

Multiplying a matrix by another matrix

Matrix multiplication

Zero matrix

Identity matrix

Properties of matrix addition

Properties of matrix scalar multiplication

Properties of matrix to matrix multiplication

Properties of matrix multiplication

Inverse of Matrices

The invertible matrix theorem

The determinant of a 2 x 2 matrix

The inverse of a 2 x 2 matrix

The inverse of a 3 x 3 matrices with matrix row operations

2 x 2 invertible matrix

Solving linear systems using 2 x 2 inverse matrices

Subspace of \(\Bbb{R}^n\)

Properties of subspace

Column space

Null space

Dimension and rank

Determinant of a Matrix

Properties of determinants

The determinant of a 3 x 3 matrix (General & Shortcut Method)

Solving linear systems using Cramer's Rule

Solving linear systems using Cramer"s Rule

Cramer"s Rule with 2 x 2 matrices

Eigenvalue and Eigenvectors

Eigenvalues and eigenvectors

The characteristic equation

Diagonalization

Complex eigenvalues

Discrete dynamical systems

Orthogonality and Least Squares

Inner product, length, and orthogonality

Orthogonal sets

Orthogonal projections

Least-squares problem

Applications to linear models

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Intro

Ex. 1

- Hello, welcome to this question right here. We're trying to graph these two equations here, and figure out a way they intersect each other, or not, right? 
So, let's go ahead and actually solve in terms of y equals mx plus b, because that's the easiest way to graph this. So, I'm gonna move the y over here and move the negative nine on this side to solve the y. So, I've got myself a 2x plus nine. And this one over here is just going to be, let me do a little bit work here, I'm gonna move the negative x over and I get this, divide everything by three, okay? 
So, now my second equation says negative one over three x plus two, okay? Now, if you have a grid paper, or just a piece of paper is fine, because we're just gonna do a rough sketch to figure out where they intersect each other. So, I've got a positive nine on top, one, two, three, four, five, six, seven, eight, nine. There's a point right there for the first equation, and it's going in this direction, up to over one. That's the slope right there. So, in another way, you can actually do the other direction. So, instead of going up two over one because I ran out of space, I can go down two left one, which is here. And down two left one, which is here. So, I've got myself a nice single straight graph coming down this way, okay? 
Now, if you do the second equation quickly it's actually, your y intercepts at the point y equals two, and the slope is this direction, because it's negative. So, you go down one and over three. So, you go down one and over three, so one, two, three, you're somewhere over here. Two points usually gives you a good point so we're gonna draw a line over here, draw a line over here. Sorry it's not right going straight.okay? 
We've got a little approximate point right here, which is right over here, okay? 
That's a point of intersection. Now, if you're wondering what that point is, it happens at negative three, and a positive three, okay? So, if you have a grid paper, grid paper you can actually plot them really good. And you will see that the both lines right here intersect each other at negative three, and three. Thanks for watching.

In order to determine the number of solutions to a system of linear equations, other than by finding the slope of the lines, we can also graph the equations out and look for the intersection of the lines.