Linear Algebra Basics

What is the result of multiplying a 2x3 matrix by a 3x4 matrix?

• 3x3

• 2x3

• 2x2

• 2x4

Define eigenvalues and eigenvectors in the context of linear algebra.

• Eigenvalues are vectors that represent how a linear transformation stretches or compresses a scalar. Eigenvectors are the scalars that remain in the same direction after the transformation, only scaled by the eigenvalue.

• Eigenvalues are scalars that represent the magnitude of a linear transformation. Eigenvectors are the vectors that remain in the same direction after the transformation, only scaled by the eigenvalue.

• Eigenvalues are vectors that represent the magnitude of a linear transformation. Eigenvectors are the vectors that change direction after the transformation.

• Eigenvalues are scalars that represent how a linear transformation stretches or compresses a vector. Eigenvectors are the vectors that remain in the same direction after the transformation, only scaled by the eigenvalue.

How do you calculate the determinant of a 2x2 matrix?

• ab - cd

• ba - dc

• ac - bd

• ad - bc

Explain the concept of matrix addition and provide an example.

• For example, given two matrices A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], the sum A + B = [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]].

• Matrix addition involves multiplying the matrices element-wise

• Matrix addition is only defined for square matrices

• The result of matrix addition is always a matrix with the same dimensions as the original matrices

What is the characteristic equation of a matrix?

• Determinant(Matrix - λ * Identity) = 0

• Determinant(Matrix + λ * Identity) = 0

• Eigenvalues(Matrix) = 0

• Trace(Matrix) = 0

How are eigenvectors related to eigenvalues?

• Eigenvectors are related to eigenvalues as the vectors associated with specific eigenvalues.

• Eigenvalues are the reciprocals of eigenvectors

• Eigenvectors have no relationship to eigenvalues

• Eigenvectors are always orthogonal to eigenvalues

Perform matrix multiplication: [1 2] * [3 4]

• 7

• 5

• 11

• 14

What is the trace of a matrix?

• The trace of a matrix is the sum of the elements on the main diagonal of the matrix.

• The trace of a matrix is the product of the elements on the main diagonal of the matrix.

• The trace of a matrix is the sum of all elements in the matrix.

• The trace of a matrix is the determinant of the matrix.

Discuss the importance of eigenvalues and eigenvectors in machine learning.

• Eigenvalues and eigenvectors are used for data encryption in machine learning.

• Eigenvalues and eigenvectors are important in machine learning for dimensionality reduction, feature extraction, and understanding data structure.

• Eigenvalues and eigenvectors are only relevant in theoretical aspects of machine learning.

• Eigenvalues and eigenvectors have no impact on the accuracy of machine learning models.

Solve for the eigenvalues of the matrix: [2 0] [0 3]

• [2, 2]

• [0, 3]

• [3, 2]

• [2, 3]