What is the formula for the norm of a vector \( v \)? \[ ||v|| = \sqrt{\sum v_i} \] \[ ||v|| = v^T \cdot v \] \[ ||v|| = \sqrt{v^T \cdot v} \] \[ ||v|| = \sum v_i^2 \]

\[ ||v|| = \sqrt{v^T \cdot v} \]

How is the distance between two vectors \( u \) and \( v \) calculated? \[ \text{dist}(u,v) = u \cdot v \] \[ \text{dist}(u,v) = ||u+v|| \] \[ \text{dist}(u,v) = ||u-v|| \] \[ \text{dist}(u,v) = \sqrt{u^T v} \]

\[ \text{dist}(u,v) = ||u-v|| \]

Which learning algorithm uses unlabeled data to extract patterns, often used for clustering? Semi-supervised learning algorithms Supervised learning algorithms Unsupervised learning algorithms Reinforcement learning algorithms

Unsupervised learning algorithms

What is the primary goal of a clustering algorithm? Maximize both intra-cluster and inter-cluster distances Minimize intra-cluster distances and maximize inter-cluster distances Maximize intra-cluster distances and minimize inter-cluster distances Minimize both intra-cluster and inter-cluster distances

Minimize intra-cluster distances and maximize inter-cluster distances

What measures the 'goodness' of a clustering when the ground truth is known or unknown? Jaccard index and Silhouette coefficient Rand index and Jaccard index Silhouette coefficient and Dunn index Rand index and Silhouette coefficient

Rand index and Silhouette coefficient

Comprehensive Study of Linear Algebra and Machine Learning Concepts

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Computer Science - Artificial Intelligence

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Cards in this deck(100)

What is the formula for the norm of a vector \( v \)?

How is the distance between two vectors \( u \) and \( v \) calculated?

Two vectors \( u \) and \( v \) are orthogonal if their dot product is _____.

Which type of learning algorithm uses labeled data to learn a function for regression and classification?

Which learning algorithm uses unlabeled data to extract patterns, often used for clustering?

What is the primary goal of a clustering algorithm?

Which algorithm involves picking \( k \) cluster centers randomly and iterating until convergence?

What measures the 'goodness' of a clustering when the ground truth is known or unknown?

Which algorithm labels a test point based on the labels of the \( k \) closest training samples?

What is the challenge with k-NN in high dimensions where distances become strange?

What is the difference between a parameter and a hyperparameter in a model?

How do you determine the 'best' \( k \) in a model while avoiding overfitting?

How can a system of \( m \) equations in \( n \) unknowns be written as a vector equation?

How is a system of \( m \) equations in \( n \) unknowns represented as a matrix equation?

What are the possible numbers of solutions for a linear system?

A vector equation is consistent if and only if vector \( b \) is in the span of _____.

A consistent system has exactly one solution if and only if the vectors form a _____.

How many floating-point operations (flops) does it take to perform Gaussian elimination on an \( n \times n \) matrix?

In Reduced Row Echelon Form (RREF), a linear system is consistent if there are no rows containing _____.

In Reduced Row Echelon Form (RREF), a linear system is unique if there are no _____.

Matrices can be thought of as functions or transformations over space. What does \( Ax = b \) represent?

What is the result of the linear transformation \( T(cu + dv) \)?

Which of the following is true if a matrix \( A \) is invertible?

Is a non-square matrix invertible?

What pattern do the first non-zero entries in each row form in Row Echelon Form?

In Row Echelon Form, what are pivot positions and pivot columns?

What are the domain, codomain, and range of a linear transformation from \( R^2 \) to \( R^3 \)?

A transformation is linear if it is closed under addition and scalar multiplication. What does this mean?

What is a one-to-one transformation in terms of mapping \( T: R^n \to R^m \)?

What does it mean for a transformation \( T: R^n \to R^m \) to be onto \( R^m \)?

If a matrix is invertible, it is both _____ and _____.

What does the determinant of a square matrix tell us?

What is the inverse of a 2x2 matrix \( A \) with elements \( a, b, c, d \)?

How do you find the inverse of an \( m \times m \) matrix larger than 2x2?

How do you find matrices for transformations such as \( T(x) = rx \) where \( r = 2.5 \)?

What are the three properties of a subspace \( H \) in \( R^n \)?

True or False: Every span is a subspace.

What is the span of a set of vectors?

What is the column space of a matrix \( A \)?

The column space of an \( m \times n \) matrix is a subspace of _____.

What is the null space of a matrix \( A \)?

The null space of an \( m \times n \) matrix is a subspace of _____.

What is the basis for a subspace \( H \) of \( R^n \)?

How do you solve for the basis of a null space?

How do you solve for the basis of a column space?

How do you find coordinates in a new basis, for example, plotting \( x=[1, 6] \) in the standard basis and relative to new basis \( B \)?

What are two common types of matrix factorization?

What does LU decomposition represent for a matrix \( A \)?

What is the \( U \) matrix in LU decomposition?

What is the \( L \) matrix in LU decomposition?

How do you solve \( Ax = b \) using LU decomposition?

What does QR decomposition represent for a matrix \( A \)?

What is the \( Q \) matrix in QR decomposition?

What is the \( R \) matrix in QR decomposition?

How do you solve \( A = QR \) in QR decomposition?

What condition must be met for vectors \( A \) and \( B \) to be orthogonal?

What is the orthogonal projection of \( y \) onto \( u \)?

What is the Gram-Schmidt orthogonalization process?

What does it mean for vectors to be orthonormal?

If \( U \) is square and has orthonormal columns, what is true about \( U \)?

What is the least squares problem in terms of minimizing \( ||Ax - b||^2 \)?

What are the normal equations for \( Ax = b \)?

If \( A^T A \) is not invertible, how many solutions will there be?

What is the basic regression task in terms of estimating parameters?

What is the design matrix \( X \) in regression analysis?

What is the observation vector \( y \) in regression analysis?

What is the parameter vector \( \beta \) in regression analysis?

What does \( y_j \) represent in regression analysis?

What does \( B_0 + B_1 x_j \) represent in regression analysis?

What is a residual in regression analysis?

How do you solve linear regression problems using the least squares method?

What is multiple regression in terms of variables?

When is a matrix \( A \) invertible in terms of computational complexity?

What is a Markov chain in terms of a dynamical system?

What is an eigenvector in terms of a square matrix \( A \)?

What are eigenvalues in terms of eigenvectors?

What is an eigenspace in terms of eigenvalues?

How do you find all eigenvalues of a matrix \( A \)?

What is a diagonal matrix?

What is the diagonalization (eigendecomposition) of a square matrix \( A \)?

An \( n \times n \) matrix is diagonalizable if and only if it has _____.

What is the four-step process to diagonalize a matrix?

What does the operation \( Ax = PDP^{-1} \) decompose into?

What are the properties of symmetric matrices?

When is \( Q \) in \( Q(x) = x^T A x \) called positive definite and positive semidefinite?

What are the advantages of basis in relation to eigenvalues and solutions?

When is \( Q \) in \( Q(x) = x^T A x \) called indefinite?

What is a stochastic matrix?

What is a steady-state vector in a Markov chain?

How do you calculate the steady-state vector in a Markov chain?

What is the largest eigenvalue of a Markov Chain?

What does the second largest eigenvalue tell us in a Markov chain?

When does a Markov chain have a unique steady-state vector?

What is the PageRank algorithm used for?

In the context of PageRank, what do vertices and edges correspond to?

How do you compute the exact PageRank result?

What is Singular Value Decomposition (SVD) in terms of matrix factorization?

What are the three steps in the linear transformation corresponding to \( A \) in SVD?

What are singular values in terms of eigenvalues?

What does the SVD theorem state about a matrix \( A \)?

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Comprehensive Study of Linear Algebra and Machine Learning Concepts

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