Linear Algebra Meets Geometry

Question 1

What is the definition of a point?

Answer 1

A location relative to the origin.

Question 2

What is the definition of a Vector?

Answer 2

Points from one point to another; comprised of direction * norm.

Question 3

What is the format of a vector?

Answer 3

The format of a Vector is the displacement needed to get to a point assuming the vector begins at the origin.

Question 4

What is the Vector pointing FROM point (1, 2) TO point (-2, -1)?

Answer 4

(-3, -3)

Question 5

How is vector addition performed?

Answer 5

Assumes same size, performed element wise.

Question 6

Does order matter in Vector addition?

Answer 6

No, it is commutative.

Question 7

What is the result of a Vector [v1, v2, v3, …, vk] multiplied by scalar s?

Answer 7

[sv1, sv2, …, s*vk]

Question 8

What is the norm of a Vector [v1, v2, v3, …, vk]?

Answer 8

sqrt(v12 + v22 +… + vk**2)

Question 9

What is the norm of a Vector?

Answer 9

It is the length or magnitude

Question 10

What is a unit vector

Answer 10

A vector with a norm equal to 1

Question 11

What is the shorthand for a vector u that is a unit vector?

Answer 11

û

Question 12

Given a vector u, how do you transform it into a unit vector?

Answer 12

û = u / (||u||) or vector divided by its norm

Question 13

Given vectors u and v, what is the dot product of both assuming they are the same size?

Answer 13

It is the sum of elementwise multiplication between the two vectors.

Question 14

Given vectors u and v, what is the numpy method to find the dot product?

Answer 14

u.dot(v)

Question 15

Given vectors u and v, what is the numpy operator to find the dot product?

Answer 15

u @ v

Question 16

Given numpy arrays u and v, where:

u = [1, 0, 1]
v = [0, -1, 2]

what does list(zip(u, v)) return?

Answer 16

[(1, 0), (0, -1), (1, 2)]

Question 17

What is the dot product of a vector v with itself?

Answer 17

||v||**2

Question 18

What is the dot product of two unit vectors?

Answer 18

cos(theta), where theta is the angle between the two vectors

Question 19

What are orthogonal vectors?

Answer 19

Non-zero vectors for which the dot product is 0, meaning the angle between them is 90 degrees.

Question 20

Cosine Similarity: What is the similarity of two vectors when cos(theta) = 1?

Answer 20

They point in the same direction

Question 21

Cosine Similarity: What is the similarity of two vectors when cos(theta) = -1?

Answer 21

They point in opposite directions

Question 22

Cosine Similarity: What is the similarity of two vectors when cos(theta) = 0?

Answer 22

They are orthogonal.

Question 23

Why is cosine similarity better than dot product for comparing vectors?

Answer 23

The norm can muddy the water, and cosine only compares direction.

Question 24

What is a linear Classifier?

Question 25

What is a Hyperplane?

Answer 25

The equivalent of a line or plane in greater than 3 dimensions.

Question 26

What is the Hyperplane defined by the equation wTu = 0?

Question 27

What is the equation for a hyper plane?

Answer 27

wTu + b= 0

Question 28

Given hyperplane wTu + b= 0, what is w?

Answer 28

The fixed weight vector

Question 29

Given hyperplane wTu + b= 0, what is b?

Answer 29

b is a scalar called the bias.

Question 30

When the bias is 0, what point will a hyperplane always pass through?

Answer 30

The origin.

Question 31

What can you say about the angle between vectors that have positive coefficients?

Answer 31

The angle is between 0 and 90 degrees.