Questions and Answers
Which of the following are valid linear algebra expressions? (Select all that apply)
What is the angle between the two vectors a = [1 -2 -1] and b = [-1 1 1]? (Select one)
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)
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Which of the following statements are generally true about transposition? (Select all that apply)
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What is the rank of the matrix representing the DataFrame given 5 duplicate rows and 5 duplicate feature vectors?
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What is the max rank of a r x c matrix if r > c?
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What is the max rank of a r x c matrix if r < c?
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What questions can be answered using linear algebra?
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In a n x d matrix, how many observations and features are there?
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Are observations row or column vectors?
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Are features row or column vectors?
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What are the three types of computing?
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What is the dot product of the two vectors v1 = (x1, y1) and v2 = (x2, y2)?
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Two vectors do not have to be the same size when taking the dot product.
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What does cos(θ) equal in a perfect correlation (dependent)?
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What does cos(θ) equal when there is no correlation (independent)?
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What is the product of the following matrix vector multiplication | a b c | | x | | d e f | | y | | g h i | | z |?
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What happens geometrically when you multiply a vector by a matrix?
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What happens geometrically when you multiply a matrix by a matrix?
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What does changing the alpha value affect?
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How do you take the transpose of a matrix?
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What is A * inv(A) equal if A is a matrix?
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What is the only type of matrix that has an inverse?
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What are vector spaces?
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What is a basis for a vector space?
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Are these vectors linear independent? v = [1 2 3], u = [0 1 2], w = [2 5 8]
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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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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.