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Linear Algebra – Vector Algebra

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    Study Guide Linear Algebra – Vector Algebra 1. The Space (R 2 ) 1 . Understanding the Coordinate Plane When you think about the usual x – y plane , you’re really thinking about something called (R 2 ) (read as “R two”). Let’s break that down in a simple way. The x – y plane is made up of all possible ordered pairs of real numbers : Each pair represents one point in the plane. To visualize this, imagine: • One horizontal number line (the x - axis) • One vertical number line (the y - axis) These two lines cross at their starting point, called the origin (0, 0) . Together, they form the coordinate axes . How Coordinates Work If you are given a point ( x 1 , x 2 ) : • The first coordinate (x 1 ) tells you how far to move left or right (horizontal movement). • The second coordinate (x 2 ) tells you how far to move up or down (vertical movement). Order matters! The point (x 1 , x 2 ) is usually different from ( x 2 , x 1 ) . That’s why we call them ordered pairs .

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    Study Guide Because every point in the plane needs two real numbers , we call this space: It means “the set of all ordered pairs of real numbers.” It is also called 2 - space . Figure 1 2. Algebraic Structure of (R 2 ) Now here’s something powerful: (R 2 ) is not just a picture of points — it also has algebraic operations . There are two main operations: 1. Addition 2. Scalar multiplication Let’s understand each one. 1. Adding Two Points Suppose we have: To add them, we simply add corresponding coordinates:

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    Study Guide So we: • Add the first numbers together • Add the second numbers together That’s it! 2. Scalar Multiplication A scalar is just a real number (like 2, −3, ½, etc.). If we multiply a point by a scalar (c), we multiply each coordinate: So the whole point stretches, shrinks, or flips depending on (c). Example 1 : Let x = (1, 3) and y = (−2, 5). Determined the points x + y, 3x, and 2x − y. Let: We’ll find: • (x + y) • (3x) • (2x - y) Step 1: Add the Points Add coordinate - wise:

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    Study Guide Step 2: Multiply by a Vector by 3 Multiply both coordinates: Step 3: Compute (2x - y) First, understand subtraction. We define subtraction as: So: Now calculate: Then subtract: So,

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    Study Guide Vectors in (R 2 ) Now let’s talk about vectors . A geometric vector is a directed line segment . It has: • An initial point (called the tail) • A terminal point (called the tip) It is usually drawn as an arrow. Figure 2 Vectors from One Point to Another Suppose we have two points: The vector from a to b is written as: To find it, subtract coordinates: These two numbers are called the components of the vector. • (b 1 - a 1 ) tells you how far you move horizontally.

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    Study Guide • (b 2 - a 2 ) tells you how far you move vertically. So the vector simply measures the movement needed to go from point a to point b . Figure 3 Example 2: Finding the Vector from One Point to Another Let’s work with two points: We want to find the vector from point a to point b , written as: Find the Components To find a vector from one point to another, we subtract coordinates: Horizontal component: Vertical component: So the vector is:

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    Study Guide This means: • Move 9 units left • Move 4 units up Figure 4 Example 3: Finding the Terminal Point of a Vector Now let’s reverse the situation. Suppose we know: • The initial point is (x = ( - 3, 5 ) • The vector is xy = (8, - 7 ) We want to find the terminal point (y) . Step 1: Add the Vector to the Initial Point To find the endpoint, add the vector components to the coordinates of the starting point. So the terminal point is:

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    Study Guide This means: • Move 8 units to the right • Move 7 units down Figure 5 Equivalent Vectors in (R 2 ) Now let’s explore an important idea. Two vectors in (R 2 ) are called equivalent (or equal) if: • They have the same first component • They have the same second component Their starting points do not matter — only the components matter. Example of Equivalent Vectors Consider the points:

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    Study Guide Step 1: Find Vector ( ab ) Horizontal component: Vertical component: So: Step 2: Find Vector ( cd) Horizontal component: Vertical component: So: Since both vectors have the same components:

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    Study Guide Even though they start in different places, they represent the same movement . Figure 6 What Does It Mean to Translate a Vector? To translate a vector means to slide it to a new position without changing its direction or length . Important idea: • The starting and ending points change • The components stay exactly the same For example: If we slide vector (ab) so that it starts at point (c = (1, - 2 ) , it will line up exactly with (cd). That’s another way to understand why: Vectors are equal if their components are equal — regardless of location. Example 4: Are These Two Vectors Equivalent? Let’s compare two vectors. First vector:

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