Exploring Linear Algebra and Machine Learning

What is a vector space?

• A vector space is a type of geometric shape.

• A vector space is a collection of points in a plane.

• A vector space is a set of numbers that cannot be added together.

• A vector space is a set of vectors that can be added together and multiplied by scalars, satisfying specific axioms.

Define a normed space.

• A normed space is a vector space with a norm that measures the size of vectors.

• A normed space is a collection of scalar values.

• A normed space is a type of matrix.

• A normed space is a geometric shape in Euclidean space.

Explain the Pythagorean Theorem in the context of vector spaces.

• The Pythagorean Theorem applies only to triangles, not vectors.

• The Pythagorean Theorem in vector spaces relates the lengths of orthogonal vectors and their resultant vector.

• The theorem states that the sum of the angles in a vector space is always 180 degrees.

• The Pythagorean Theorem is used to calculate the area of a vector space.

What are the properties of inner product spaces?

• Orthogonality, continuity, compactness

• Completeness, boundedness, convergence

• Distributivity, associativity, closure

• Linearity, symmetry, positive definiteness

Define eigenvalues and eigenvectors.

• Eigenvalues are the dimensions of a matrix; eigenvectors are the rows of the matrix.

• Eigenvalues are scalars indicating the factor by which an eigenvector is scaled during a transformation; eigenvectors are the vectors that are scaled by the transformation.

• Eigenvalues are always positive numbers; eigenvectors are always unit vectors.

• Eigenvalues are the angles of rotation; eigenvectors are the axes of rotation.

How do you find the eigenvalues of a matrix?

• Solve det(A - λI) = 0 for λ.

• Add the matrix to its transpose.

• Find the inverse of the matrix.

• Calculate the trace of the matrix.

What is the Fundamental Theorem of Linear Algebra?

• The rank-nullity theorem and the relationship between row space and column space.

• The concept of limits in calculus.

• The relationship between eigenvalues and eigenvectors.

• The Pythagorean theorem and its applications in geometry.

Explain the concept of orthogonality in vector spaces.

• Orthogonality refers to the length of a vector.

• Orthogonality is the concept of vectors being equal in magnitude.

• Orthogonality means that two vectors are perpendicular, having a dot product of zero.

• Orthogonality means that two vectors are parallel, having a dot product of one.

What are the different types of machine learning?

• Supervised analysis

• Unsupervised classification

• Reinforcement prediction

• Supervised learning, Unsupervised learning, Reinforcement learning

What is unsupervised learning?

• A method that requires labeled data for training.

• A technique used only for classification tasks.

• A process that eliminates all data noise before analysis.

• Unsupervised learning is a machine learning approach that finds patterns in data without labeled outputs.