24 June 2025

#Linear Algebra

#Linear Algebra
What is a scalar?
Define a vector.
What is a matrix?
What is a tensor?
How do you represent a vector geometrically?
What is the difference between a row vector and a column vector?
What is a diagonal matrix?
What is an identity matrix?
What is a zero matrix?
What is a symmetric matrix?
What is a skew-symmetric matrix?
Define orthogonal matrix.
What is a triangular matrix?
What is the trace of a matrix?
What does it mean for a matrix to be invertible?
What is matrix addition?
What is scalar multiplication?
Define matrix multiplication.
When is matrix multiplication possible?
Is matrix multiplication commutative?
What is the associative property of matrix multiplication?
What is the distributive property?
What is the transpose of a matrix?
What are the properties of matrix transpose?
What is the inverse of a matrix?
What are the conditions for a matrix to have an inverse?
What is a singular matrix?
What is an orthogonal matrix?
How do you compute the determinant of a 2x2 matrix?
How do you compute the determinant of a 3x3 matrix?
What is the dot product of two vectors?
What is the cross product?
What does it mean for two vectors to be orthogonal?
What is the angle between two vectors?
What is the projection of one vector onto another?
What is a unit vector?
What is a basis vector?
How do you normalize a vector?
What is vector space?
What is subspace?
What is linear independence?
What is linear dependence?
How do you check if vectors are linearly independent?
What is the span of a set of vectors?
What is the basis of a vector space?
What is the dimension of a vector space?
Give an example of a dependent set of vectors.
What is the rank of a matrix?
How do you compute the rank?
What does full rank mean?
What is the null space (kernel) of a matrix?
How do you find the nullity of a matrix?
What is the column space?
What is the row space?
What is the orthogonal complement?
What is an eigenvalue?
What is an eigenvector?
How do you compute eigenvalues?
What is the characteristic polynomial?
How do you find eigenvectors?
What is geometric multiplicity?
What is algebraic multiplicity?
What are some properties of eigenvalues?
Why are eigenvalues important in machine learning?
What is the spectral theorem?
What is LU decomposition?
What is QR decomposition?
What is SVD (Singular Value Decomposition)?
What is the purpose of matrix decomposition?
What is the Cholesky decomposition?
What is the difference between LU and QR?
What is the Gram-Schmidt process?
How is SVD used in data compression?
What is the Jordan Normal Form?
What is the Schur decomposition?
What is the determinant of a matrix?
What is the physical meaning of a determinant?
How do you find the inverse of a 2x2 matrix?
How do you find the inverse using row reduction?
What is the adjoint of a matrix?
How is the determinant used to check invertibility?
What is a cofactor?
What is the Laplace expansion?
Can a non-square matrix be inverted?
What is a pseudoinverse?
What is the Moore-Penrose pseudoinverse?
What is a block matrix?
What is the Kronecker product?
What is the Hadamard product?
What is matrix exponentiation?
What is a condition number?
What is a rank-deficient matrix?
What is a positive definite matrix?
What is a diagonalizable matrix?
What is matrix similarity?
What is a vector norm?
What is the L1 norm?
What is the L2 norm?
What is the infinity norm?
What is the Frobenius norm?
How are norms used in machine learning?
What is the distance between two vectors?
How is cosine similarity calculated?
What is the relationship between norm and distance?
How does normalization affect vector norms?
What is an orthogonal set of vectors?
What is orthonormality?
How do you orthogonalize a set of vectors?
What is the projection matrix?
How do you project a vector onto a subspace?
What is the geometric interpretation of a projection?
When is a projection matrix idempotent?
What is the Gram matrix?
What is the orthogonal projection theorem?
How is projection used in least squares?
What is the least squares solution?
Why is least squares used in linear regression?
What is the normal equation?
How do you derive the least squares estimator?
What happens if the design matrix is not full rank?
What is the role of the pseudoinverse in least squares?
What is the residual vector?
How do you minimize the residual?
What is the cost function in linear regression?
What is the relationship between projection and least squares?
What is a linear transformation?
What is the matrix representation of a transformation?
What is a standard basis?
What is a change of basis?
How do you change a vector to a new basis?
What is a transition matrix?
How are linear transformations represented in different bases?
What is the role of similarity transformations?
What is the canonical form?
What is the matrix of a reflection or rotation?
What is a symmetric matrix?
What is a skew-symmetric matrix?
What is a positive definite matrix?
What is a positive semi-definite matrix?
How do you test for positive definiteness?
What are the properties of symmetric matrices?
What are applications of positive definite matrices in ML?
What is an idempotent matrix?
What is a nilpotent matrix?
What is the Cayley-Hamilton Theorem?
What is Gaussian elimination?
What is Gauss-Jordan elimination?
What is row echelon form?
What is reduced row echelon form?
What is pivoting?
What are leading and free variables?
What is backward substitution?
What is forward substitution?
What is the computational complexity of matrix multiplication?
What is sparse matrix representation?
What is SVD used for in NLP?
How is PCA related to eigenvectors?
What is the Eckart?Young theorem?
What is the thin SVD?
What is truncated SVD?
What is the application of QR decomposition in ML?
How does Cholesky compare to LU?
What is the Householder transformation?
What is the Givens rotation?
What is the role of decomposition in solving linear systems?
What is a dual space?
What is a linear functional?
What is the relationship between dual basis and basis?
How are linear maps between duals represented?
What is reflexivity in linear algebra?
How are eigenvectors used in spectral clustering?
What is Laplacian matrix in graph theory?
What are principal components?
How is PCA implemented using eigen decomposition?
How does dimensionality reduction work in PCA?
What is a vector subspace?
What is the annihilator of a subspace?
What is the quotient space?
What is the rank-nullity theorem?
What is a bilinear form?
What is a quadratic form?
How do you diagonalize a quadratic form?
What is matrix congruence?
What is orthogonal diagonalization?
What is a linear operator?
Why is linear algebra essential in machine learning?
How is linear algebra used in neural networks?
What is the role of dot product in attention mechanisms?
How are matrices used in image processing?
What is the Jacobian matrix in deep learning?
What is the Hessian matrix?
How is SVD used in recommendation systems?
How are tensors used in deep learning?
What is the shape of input data for neural networks?
How does dimensionality reduction improve performance?
What is a tensor?
What is a rank of a tensor?
How are tensors represented?
What are tensor contractions?
What is tensor decomposition?
Can a matrix have more than one inverse?
Can a non-square matrix be orthogonal?
Can a set of linearly independent vectors form a basis?
Is every orthonormal set linearly independent?
Can two vectors be orthogonal but not linearly independent?
What is the geometric meaning of a determinant?
How is the rank related to dimensionality?
How do transformations affect shapes and dimensions?
What is shearing?
What is scaling in linear transformations?
Difference between rank and dimension?
Compare dot product and cross product.
Difference between linear map and affine map?
Compare eigen decomposition and SVD.
Difference between orthogonal and orthonormal?
True or False: All orthogonal matrices are invertible.
True or False: A matrix with zero determinant is invertible.
True or False: Eigenvalues can be complex.
True or False: A matrix can be diagonalized only if it's square.
True or False: The transpose of a symmetric matrix is symmetric.
Describe how a matrix transforms a vector.
Explain the intuition behind the dot product.
Describe how matrix rank affects system solutions.
Explain the steps to solve a linear system using Gaussian elimination.
Describe why PCA reduces noise.
How do you compute matrix inverse in Python (NumPy)?
How do you compute eigenvalues in NumPy?
How do you use SVD in scikit-learn?
How do you create a projection matrix using NumPy?
How do you find the rank of a matrix programmatically?
Given 3 vectors, check if they are linearly independent.
Find the eigenvalues of a 2x2 matrix manually.
Solve a 3x3 linear system using matrix inversion.
Given a matrix, compute its rank.
Reduce a matrix to row echelon form.
What is the determinant of an identity matrix?
What is the transpose of a diagonal matrix?
What is the rank of a zero matrix?
What is the nullity of an invertible matrix?
What is the dimension of R³?
Why are orthogonal matrices preferred in numerical computations?
Why is SVD preferred over eigen decomposition in some cases?
How is linear algebra related to convolution in CNNs?
How do singular matrices affect training in ML models?
What matrix operations are most common in backpropagation?
What is a Vandermonde matrix?
What is the companion matrix?
What is the Toeplitz matrix?
What is a circulant matrix?
What are real-world examples of linear algebra in AI?
  • Scalars, Vectors, Matrices, and Tensors
  • Matrix Operations
  • Dot Product (Inner Product)
  • Matrix Multiplication
  • Transpose of a Matrix
  • Identity Matrix (I)
  • Inverse Matrix
  • Determinant
  • Rank of a Matrix
  • Linear Independence
  • Orthogonality
  • Eigenvalues and Eigenvectors
  • Singular Value Decomposition (SVD)
  • Norms (Vector Magnitude)
  • Projection
  • Row Space and Column Space
  • Null Space (Kernel)
  • Basis and Dimension
  • Diagonalization of Matrices
  • Trace of a Matrix
  • Symmetric Matrices
  • Skew-Symmetric Matrices
  • Orthogonal Matrices
  • Positive Definite Matrices
  • LU Decomposition
  • QR Decomposition
  • Cholesky Decomposition
  • Rank-Deficient Matrices
  • Gram-Schmidt Process
  • Moore-Penrose Pseudoinverse
  • Condition Number of a Matrix
  • Matrix Norms
  • Block Matrices
  • Sparse Matrices
  • Band Matrices
  • Triangular Matrices (Upper and Lower)
  • Permutation Matrices
  • Idempotent Matrices
  • Nilpotent Matrices
  • Jordan Normal Form
  • Cayley-Hamilton Theorem
  • Schur Decomposition
  • Householder Transformation
  • Givens Rotation
  • Rank-One Update
  • Outer Product of Vectors
  • Kronecker Product
  • Hadamard Product
  • Matrix Exponential

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