Introduction to Linear Algebra

Question 1

What is a matrix?

Answer 1

A matrix is a collection of numbers ordered in rows and columns

Question 2

What is each of the values of a matrix calle?

Answer 2

an element

Question 3

In Linear Algebra what are the two dimensions of the matrix?

Answer 3

Rows and Columns

Example matrix A with 2 rows and 3 columns. We say its dimensions are 2x3
Or
A is a 2 by 3 matrix

Question 4

What is the last element in a matrix?

Answer 4

a mn

Question 5

How do you calculate the total number of elements

Answer 5

m*n

Question 6

A matrix can contain:

Numbers

Symbols

Expressions

All of the above

Answer 6

All of the above

Question 7

How many elements does a matrix with dimensions m = 5 and n = 12 have?

5

12

60

120

Answer 7

60

Question 8

What is a vector?

Answer 8

A vector is practically the simplest linear algebraic object

They sit somewhere between scalar and matrix as they have 1 dimension

ie. a 3 by 1 matrix

a matrix can be viewed a collection of vectors

Question 9

What is a scalar?

Answer 9

A matrix with 1 row and 1 column

0 dimensions

Question 10

What are the two types of vectors?

Answer 10

Row Vectors
Column Vectors

Question 11

What is the length of a vector?

Answer 11

The number of elements in a vector

Question 12

Choose the correct option. All numbers known from algebra are referred to as _________ in linear algebra.

Scalars

Vectors

Matrices

Tensors

Answer 12

Scalars

Question 13

If a vector has a length of 3, it has:

3 rows

3 columns

3 rows and 3 columns

3 rows or 3 columns

Answer 13

3 rows or 3 columns

Question 14

How many dimensions does a vector have?

0

1

2

3

Answer 14

1

Question 15

If a scalar has 0 dimensions, what else has zero dimensions?

Answer 15

geometrical idea of a point
- no direction or size

Question 16

In terms of geometry a scalar can be represented as:

a line

a square

a point

it cannot be represented geometrically

Answer 16

a point

Question 17

Geometrically a vector can be represented as:

A point

A line

A plane

Answer 17

A line

Question 18

A point

A line

A plane

Answer 18

A plane

Question 19

in python, what kind of vector is default?

Answer 19

a row vector

ie. an array

Question 20

What are tensors?

Answer 20

Tensors are a generalization of the concepts we have seen so far:

name - tensor rank
scalar = rank 0
vector = rank 1
matrix = rank 2
a collection of matrices = rank 3

Question 21

What is a Rank 3 tensor?

Answer 21

A collection of matrices
Dimensions of k x m x n

Question 22

Choose the best answer:

Scalars are type of tensors

Tensors are types of matrices

Matrices are types of vectors

it cannot be represented geometrically

Answer 22

Scalars are type of tensors

Question 23

A vector is a tensor by definition. Of what rank?

0

1

2

3

Other

Answer 23

Rank1

Question 24

What happens when we transpose the same vector twice?

Answer 24

It yields the same vector

Question 25

What does it mean to transpose a vector?

Answer 25

To change it from a linear to horizontal or vice versa rows to columns and vice versa ie. 1x3 to 3x1

Question 26

What is the difference between a dot product and tensor product?

Question 27

What is another name for the dot product?

Answer 27

inner or scalar product because we multiplied two vectors and got a scalar ie. multiply two arrays and get an int The sum of the products of the corresponding elements

Question 28

What is the notation of the dot product?

Answer 28

a dot - "." - signifying it is the dot product

Question 29

What happens when we multiply a scalar by a vector?

Answer 29

we get a vector with the same length

Question 30

What is the outcome of the dot product of the following two matrices? A. B = 2x3 3x6

Answer 30

A. B C 2x3 3x6 = 2x6

Question 31

T or F - we always multiply a row vector by a column vector

Answer 31

T

Question 32

What are practical applications of linear algebra in data science?

Answer 32

vectorizing code - input matrix - weights matrix(coefficients) - output matrix imagine recognition dimensionality reduction

Question 33

What is is called when we use linear algebra to compute many values simultaneously?

Answer 33

Array programming or vectorizing code