Introduction to Linear Algebra Flashcards
What is a matrix?
A matrix is a collection of numbers ordered in rows and columns
What is each of the values of a matrix calle?
an element
In Linear Algebra what are the two dimensions of the matrix?
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
What is the last element in a matrix?
a mn
How do you calculate the total number of elements
m*n
A matrix can contain:
Numbers
Symbols
Expressions
All of the above
All of the above
How many elements does a matrix with dimensions m = 5 and n = 12 have?
5
12
60
120
60
What is a vector?
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
What is a scalar?
A matrix with 1 row and 1 column
0 dimensions
What are the two types of vectors?
Row Vectors
Column Vectors
What is the length of a vector?
The number of elements in a vector
Choose the correct option. All numbers known from algebra are referred to as _________ in linear algebra.
Scalars
Vectors
Matrices
Tensors
Scalars
If a vector has a length of 3, it has:
3 rows
3 columns
3 rows and 3 columns
3 rows or 3 columns
3 rows or 3 columns
How many dimensions does a vector have?
0
1
2
3
1
If a scalar has 0 dimensions, what else has zero dimensions?
geometrical idea of a point
- no direction or size
In terms of geometry a scalar can be represented as:
a line
a square
a point
it cannot be represented geometrically
a point
Geometrically a vector can be represented as:
A point
A line
A plane
A line
A point
A line
A plane
A plane
in python, what kind of vector is default?
a row vector
ie. an array
What are tensors?
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
What is a Rank 3 tensor?
A collection of matrices
Dimensions of k x m x n
Choose the best answer:
Scalars are type of tensors
Tensors are types of matrices
Matrices are types of vectors
it cannot be represented geometrically
Scalars are type of tensors
A vector is a tensor by definition. Of what rank?
0
1
2
3
Other
Rank1
What happens when we transpose the same vector twice?
It yields the same vector
What does it mean to transpose a vector?
To change it from a linear to horizontal or vice versa
rows to columns and vice versa
ie. 1x3 to 3x1
What is the difference between a dot product and tensor product?
What is another name for the dot product?
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
What is the notation of the dot product?
a dot - “.” - signifying it is the dot product
What happens when we multiply a scalar by a vector?
we get a vector with the same length
What is the outcome of the dot product of the following two matrices?
A. B =
2x3 3x6
A. B C
2x3 3x6 = 2x6
T or F - we always multiply a row vector by a column vector
T
What are practical applications of linear algebra in data science?
vectorizing code
- input matrix - weights matrix(coefficients) - output matrix
imagine recognition
dimensionality reduction
What is is called when we use linear algebra to compute many values simultaneously?
Array programming or vectorizing code
