Q1. Scenario: Solve the linear system 3x + 2y = 5, x + y = 2 using numpy.linalg.solve.
A = np.array([[3,2],[1,1]]); b = np.array([5,2]); solution = np.linalg.solve(A,b); returns [1.,1.] (x=1,y=1). Check: 3*1+2*1=5, 1+1=2. This is used for normal equations in linear regression.
Q2. Scenario: Compute the inverse of a matrix A = [[4,7],[2,6]] using np.linalg.inv. Then verify that A * inv(A) = identity.
A = np.array([[4,7],[2,6]]); invA = np.linalg.inv(A); product = A @ invA; should be approx identity. invA = [[0.6, -0.7],[-0.2,0.4]]. product = [[1,0],[0,1]]. Inverses used in many algorithms but computationally expensive.
Q3. Scenario: Perform matrix multiplication between a 3x2 matrix and a 2x4 matrix using np.dot, @ operator, and .matmul method.
A = np.random.rand(3,2); B = np.random.rand(2,4); C1 = np.dot(A,B); C2 = A @ B; C3 = A.matmul(B) (if using np.matmul). All produce (3,4). @ is recommended for readability. Matrix multiplication is core to neural network forward pass.
Q4. Scenario: Compute the eigenvalues and eigenvectors of matrix [[4,1],[2,3]] using np.linalg.eig.
A = np.array([[4,1],[2,3]]); eigvals, eigvecs = np.linalg.eig(A); eigvals ~ [5.,2.]; eigvecs = [[0.7071, -0.4472],[0.7071, 0.8944]]. Eigen decomposition is used in PCA and dimensionality reduction.
Q5. Scenario: Compute the L2 norm (Euclidean norm) of a vector [3,4] using np.linalg.norm. Also compute the Frobenius norm of a 2x2 matrix.
vec = np.array([3,4]); norm_vec = np.linalg.norm(vec) -> 5.0. mat = np.array([[1,2],[3,4]]); norm_mat = np.linalg.norm(mat) -> sqrt(1+4+9+16)=~5.477. Norms are used for regularization (L2 penalty).