Linear Algebra: Vectors and Systems of Equations Basics

The three possible solutions for a system are:

- no solution - infinite solutions - a unique solution

When are Vectors equal?

Same magnitude Same direction

How are vectors added and subtracted ALGEBRAICALLY (given just components)?

Simply add/subtract their corresponding x, y, z components to obtain the resultant vector.

Solution Set

The set of all possible n-tuple solutions to a system.

What are the 3 main ways to view vectors

1. A magnitude with direction (graphically) 2. Variables in an equation 3. An ordered list of numbers corresponding to respective directions

Why does a vector represent an infinite number of lines?

A vector is simply a magnitude with a direction. It can be represented in numerous locations since many different points give us the same magnitude and direction.

Position Vector?

A vector with its initial point at the origin of the graph. A = {0,0,0} B = {a₁,a₂, a₃} Vector AB is thus simplified to: {a₁,a₂,a₃}

Graphically: What does adding/subtracting vectors do?

Addition and subtraction are what add directional aspects to the vector (assuming the vectors are not colinear) In other words, addition/subtraction move us off the initial vector line.

What is the unit vector of vector a and how is it found?

Also known as the direction vector since it is used to isolate direction of the vector. It points in the same direction as v but has a magnitude of 1. A vector with a terminal point on the unit circle. Divide v's components by the magnitude of v to get it.

What is a free variable?

An additional variable that can take on any value of choice for a solution to a system. The other variables are defined in terms of the free variable. Thus making them parametric equations.

Linear Equation

An algebraic equation where each term is either: - a constant - or a product of a constants and a degree-one variables.

How is the magnitude of a vector found?

Apply the distance formula using your x, y, and z components. This is just the pythagorean theorem.

How are vectors ADDED GRAPHICALLY? How is the resultant drawn?

Attach the head of A to the tail of B, then attach the head of B to the tail of C and so on... Resultant is drawn by attaching the tail to the tail of A and the head to the head of B

How are vectors alternatively written in i, j k form?

Basically, it's instead written as a sum of each individual component scaling up its respective standard basis vector component.

For a system of equations: When is it consistent and when is it inconsistent?

Consistent: if the system has at least one solution Inconsistent: if there's no solution.

Standard basis vectors?

Each points towards its respective axis. ith = <1, 0, 0> jth = <0, 1, 0> kth = <0, 0, 1>

Graphically: What does scaling (multiplying) a vector do?

For scalar a: - Extends the vector line ( a > 1) - Shortens the vector line (a < 1) - Points in the opposite direction (-a) Scalars can't change the direction except for in the complete opposite direction when the scalar is of the opposite sign.

What is the shortcut to determine if a system of equations has free variables?

If there are more variables than equations.

When are equations Independent/Dependent?

Independent: if there is a single solution. Dependent: if there are infinite solutions.

Elimination Method

Involves multiplying and adding/subtracting the systems as needed to isolate the answer for one variable. THEN substituting the answer back into the equations to find the OTHER variable to solve the entire system.

How are vectors SUBTRACTED GRAPHICALLY? How is the resultant drawn?

Similar to adding, except for the vector being subtracted you first point it in the opposite direction THEN attach tail to head. Basically like adding a negative version of the vector.

Graphically: A system has infinite solutions if:

The equations are equal. Same slope and y-intercept.

Graphically: a system of equations has no solution if?

They are parallel lines where they have the same slope but a DIFFERENT Y-INTERCEPT. They never intersect so there's no way to have points that they share to solve both equations.

When are Systems of Equations Equivalent?

They have the same solution set.

Given an angle and a magnitude (resultant) of a vector: how are the x and y components found?

This is just like finding the legs of a triangle given hypotenuse and an angle.

What is the solution to a system of equations?

a set of numbers (s₁, s₂, s₃...) that when plugged into each equation's respective variables in the system, they are simultaneously satisfied.

Graphically: a system of equations has a solution at _____

at the point of intersection. The point where lines have a common point to solve their corresponding equations simultaneously.

To say a vector is in Rⁿ means?

it has n components in the domain of all real numbers

Given two points: how can you derive vectors in component form?

terminal point - initial point add z-coordinates for 3D-space

What is another name for the tail and head of a vector?

the initial and terminal point respectively.

How is the angle between vector components found?

use arctan to isolate theta.