Vectors and Linear Combinations of Vectors

Question 1

What is a vector?

Answer 1

A vector is a quantity having direction as well as magnitude, especially as determining the position of one point in space relative to another.

Question 2

A matrix with only one column is called …

Answer 2

… a column vector, or simply vector.

Question 3

The set of all vectors with two entires is denoted …

Answer 3

… Rˆ2

Question 4

Two vectors in Rˆ2 are equal if and only if their corresponding entries are equal. True or false?

Answer 4

True

Question 5

[1 , 2] = [2 , 1]. True or false?

Answer 5

False. Vectors in Rˆ2 are ordered pairs of real numbers.

Question 6

What can you represent with a vector?

Answer 6

• document
• binary string
• collection of attributes
• state of a system
• probability distribution

Question 7

You have these vectors:
[u1, u2, …, un]
[v1, v2, …, vn]

Add them.

Answer 7

[u1, u2, …, un] + [v1, v2, …, vn] = [u1+v1,…un+vn]

Question 8

When multiplying a vector by a scalar …

Answer 8

… you multiply each entry of the vector by that scalar

Question 9

|3|
| | = | 3 -1 |
|-1|

True or false?

it’s hard to figure out what that is, but it’s a vector equals another vector

Answer 9

False, because the matrices have different shapes, even though they have the same entries

Question 10

What is the Parallelogram Rule for Addition?

Answer 10

If u and v in Rˆ2 are represented as points in the plane, then u+v corresponds to the fourth vertex of the parallelogram whose other vertices are u, 0, and v

SO, you copy the lower line of the parallelogram to the end point of the left line of the parallelogram, and then you copy the left line of the parallelogram to the end point of the lower line of the parallelogram and then you draw the diagonal. That diagonal is the sum of the two vectors.

Question 11

What happens, graphically, when you multiply a vector with a negative scalar?

Answer 11

It changes the direction of the vector’s arrow. and the magnitude changes probably, depends what scalar you are multiplying it to.

Question 12

Vectors in Rˆ3 are __x__ matrices and they are represented geometrically by points in a ____-dimensional coordinate space.

Answer 12

3x1

three-dimensional

Question 13

If n is a positive integer, Rˆn denotes the collection of …

Answer 13

… all lists (or ordered n-tuples) of n real numbers, usually written as n x 1 column matrices

Question 14

Think of the algebraic properties of vectors, name some.

Answer 14

Commutative vector
Associative vector
Distributive vector
Distributive scalar
Associative scalar
Multiplicative identity for the real number 1
Additive inverse - P (for any P such that P + (-P) = 0)

Question 15

Just a reminder that there is a thing called “Combining vector addition and scalar multiplication” and maybe you should check it out now in Lecture 9.

Answer 15

I knew you re not gonna check it :(

Question 16

What is a linear combination?

Answer 16

A linear combination is the vector y defined by
y = c1 v1 + c2 v2 + … + cp vp

with v1, v2, …. vp vectors in Rˆn and
c1, c2, …., cp scalars (called weights)

Definition in words: a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results.
The terms are vectors in our case. The constants are scalars

Question 17

Can a linear combination of some vectors and weights be 0?

Answer 17

Yes, because the weights in a linear combination can be any real numbers, including 0.

Question 18

Can you give a definition to linear combination?

Answer 18

A linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results.

Question 19

What is the dot-product?

Answer 19

The dot-product of two vectors is the sum of product of the corresponding entries.

the vectors need to have equal sizes

Question 20

The output of a dot-product is a vector. True or false?

Answer 20

False. The output of the dot-product is a scalar. That is why, the dot-product is sometimes called the scalar product.