Linear Algebra for Data Science Flashcards

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Questions and Answers

Which of the following are valid linear algebra expressions? (Select all that apply)

A + B

(ABC)^{-1} (correct)

CBx

A + BC (correct)

(AB)^{-1} (correct)

Ax + Cx

What is the angle between the two vectors a = [1 -2 -1] and b = [-1 1 1]? (Select one)

90 degrees (correct)

45 degrees

0 degrees

60 degrees

What is the formula to find the angle θ between two vectors u and v?

θ = arccos( [u * v] / [||u|| * ||v||])

Which of the following products are valid given the matrices A, B, and C with dimensions 3x5, 4x3, and 5x8 respectively? (Select all that apply)

A * C Signup and view all the answers

Which of the following statements are generally true about transposition? (Select all that apply)

(A * B)^T = B^T * A^T Signup and view all the answers

What is the rank of the matrix representing the DataFrame given 5 duplicate rows and 5 duplicate feature vectors?

n - 5 Signup and view all the answers

What is the max rank of a r x c matrix if r > c?

c Signup and view all the answers

What is the max rank of a r x c matrix if r < c?

r Signup and view all the answers

What questions can be answered using linear algebra?

Solving large systems of linear equations, computer vision, machine learning (support vector machines, principal component analysis), optimization techniques, fitting an arbitrary polynomial. Signup and view all the answers

In a n x d matrix, how many observations and features are there?

n observations, d features Signup and view all the answers

Are observations row or column vectors?

row vectors Signup and view all the answers

Are features row or column vectors?

column vectors Signup and view all the answers

What are the three types of computing?

CPU -> GPU -> TPU (tensor processing units) Signup and view all the answers

What is the dot product of the two vectors v1 = (x1, y1) and v2 = (x2, y2)?

x1x2 + y1y2 Signup and view all the answers

Two vectors do not have to be the same size when taking the dot product.

False Signup and view all the answers

What does cos(θ) equal in a perfect correlation (dependent)?

cos(θ) = 1, θ = 0 Signup and view all the answers

What does cos(θ) equal when there is no correlation (independent)?

cos(θ) = 0, θ = 90 degrees Signup and view all the answers

What is the product of the following matrix vector multiplication | a b c | | x | | d e f | | y | | g h i | | z |?

What happens geometrically when you multiply a vector by a matrix?

<ol> <li>Rotation: vector rotates around the origin (clockwise or counterclockwise). 2) Scaling: vector gets scaled & length changes.</li> </ol> Signup and view all the answers

What happens geometrically when you multiply a matrix by a matrix?

Performs many rotations and many scalings. Signup and view all the answers

What does changing the alpha value affect?

transparency (0 for transparent, 1 for solid) Signup and view all the answers

How do you take the transpose of a matrix?

Turn the rows into columns, and columns into rows. Signup and view all the answers

What is A * inv(A) equal if A is a matrix?

identity matrix I Signup and view all the answers

What is the only type of matrix that has an inverse?

square matrices (n x n) Signup and view all the answers

What are vector spaces?

A collection of vectors with operations such as addition, subtraction, and properties such as commutativity and associativity. Signup and view all the answers

What is a basis for a vector space?

A subset of the vector space where any vector in the vector space can be spanned by a linear combination of the vectors of the basis. Signup and view all the answers

Are these vectors linear independent? v = [1 2 3], u = [0 1 2], w = [2 5 8]

False Signup and view all the answers

Study Notes

Valid Linear Algebra Expressions

Matrix A is nxn, B is nxm, C is mxn with m > n, and x is a vector nx1.
Valid expressions include A+BC, (AB)⁻¹, and (ABC)⁻¹.

Angle Between Vectors

Vectors a = [1, -2, -1] and b = [-1, 1, 1] are orthogonal.
The angle between these vectors is 90 degrees.

Formula for Angle Between Vectors

Angle θ between vectors u and v can be calculated using: θ = arccos( [u * v] / [||u|| * ||v||]).

Valid Matrix Products

Given matrices A (3x5), B (4x3), and C (5x8), valid product combinations include:
- B*A,
- BAC,
- A*C.

Properties of Transpose

A^T = A indicates symmetric matrices.
(A*B)⁻¹ = B⁻¹ * A⁻¹; transpose of a product reverses the order.
(A^T)⁻¹ = A; double transposition returns the original matrix.

Rank of a DataFrame

In a DataFrame with n rows and m columns (m > n), with 5 duplicates, the rank is n - 5.

Max Rank of Matrices

For a r x c matrix:
- If r > c, max rank is c.
- If r < c, max rank is r.

Applications of Linear Algebra

Useful for solving large systems of equations, in computer vision, machine learning (like SVMs and PCA), and optimization techniques.

Observations and Features

In a n x d matrix, there are n observations and d features.
Observations are organized as row vectors.
Features are represented as column vectors.

Types of Computing

The progression in computing types: CPU (central processing unit), GPU (graphics processing unit), TPU (tensor processing unit).

Dot Product Calculation

The dot product of vectors v1 = (x1, y1) and v2 = (x2, y2) is calculated as x1x2 + y1y2.

Dot Product Size Requirement

Two vectors must be the same size to compute their dot product; otherwise, the statement is false.

Correlation and Cosine

Perfect correlation: cos(θ) = 1, leading to θ = 0 degrees.
No correlation: cos(θ) = 0, resulting in θ = 90 degrees.

Matrix-Vector Multiplication

The product of a matrix and a vector retains the structure of the result, computed as individual sums of products for each row.

Geometric Interpretation of Vector-Matrix Multiplication

Multiplying a vector by a matrix can result in rotation (around the origin) and scaling (changing vector length).

Geometric Interpretation of Matrix-Matrix Multiplication

Multiplying matrices combines multiple rotations and scalings based on their compositions.

Alpha Value in Graphics

Adjusting the alpha value changes transparency: 0 indicates full transparency, 1 indicates fully opaque.

Transposing a Matrix

Transposition involves converting rows into columns and columns into rows.

Inverse of a Matrix

The product of a matrix A and its inverse inv(A) equals the identity matrix I.

Type of Matrix with Inverse

Only square matrices (n x n) possess an inverse.

Vector Spaces

Vector spaces consist of collections of vectors capable of operations like addition and subtraction, adhering to properties like commutativity and associativity.

Basis of a Vector Space

A basis is a subset of a vector space such that any vector can be expressed as a linear combination of these basis vectors.

Linear Independence of Vectors

Vectors v = [1, 2, 3], u = [0, 1, 2], and w = [2, 5, 8] are not linearly independent, as w can be expressed as a combination of v and u.

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Description

Test your understanding of linear algebra concepts crucial for data science with these flashcards. Each card presents a question that requires you to think critically about matrix operations and vector calculations. Perfect for students looking to reinforce their knowledge in a practical way.