Linear Algebra

Question 1

• What is a vector? Give an example.
• Where in space does it start?
• Why is vector mathematical notation useful?

Answer 1

A vector is something that has both magnitude and direction.
- 5 mph is a speed.
- East is a direction.
- 5 mph east is a velocity, a vector.

We don’t necessarily care about their origin.

Mathematical notation is useful because we can represent more than 3 dimensions.

Question 2

• What is a real coordinate space?
• What is R^2 and what is it made of?
• What is a tuple?

Answer 2

• A Euclidean space of n dimensions where each point is defined by n real numbers, representing coordinates.
• R^2 is a 2-dimensional real coordinate space. It is made of all the possible real-valued 2-tuples.
• A tuple is an ordered list of numbers.

Question 3

• How do you add vectors numerically?
• Does the order in which you sum vectors matter?
• How do you add them graphically?

Answer 3

• You add vectors numerically by summing the corresponding components.
• You can add vector b to vector a or vice versa. Either way it will result in vector c.
• You add vectors graphically by placing the tail of one on the tip of the other. The tip of the second one will land on the tip of the resulting vector.

Question 4

• What is a scalar?
• How do you multiply a vector by a scalar?
• What happens to the magnitude and direction of the vector when scaled?

Answer 4

A scalar is just a magnitude. It has no direction.

You multiply the scalar by each of the components of the vector.

It simply scales up or down the vector without changing its direction, except when the scalar is negative, which flips the vector.

Question 5

• What are unit vectors?
• How many unit vectors must be defined?
• How are they represented symbolically (accent)?
• How are they used to represent all other vectors?

Answer 5

Unit vectors are vectors that move one unit in one specific direction and 0 in all others.

We must define a unit vector for every dimension in our space.

Unit vectors are represented as lower-case letters with hats or arrows.

Any n-dimensional vector we can represent as a sum of scaled versions of n unit vectors.