Math Solutions

Verify that Q = [[cos θ, −sin θ],[sin θ, cos θ]] is orthogonal by computing QᵀQ and QQᵀ. Show that Q preserves lengths and angles.

Fit a line y = mx + c through the points (1,1), (2,3), (3,2) using the normal equations AᵀAx̂ = Aᵀb. Compute m and c.

Project b = [1, 2, 3] onto the column space of A = [[1, 0],[0, 1],[0, 0]]. Show the projection and the residual.

Orthogonalize the set {[1,1,0], [1,0,1]} using the Gram–Schmidt process, then normalize to get an orthonormal basis.

Verify that u = [1, 2, −1] and v = [3, 0, 3] are orthogonal by computing u · v. Show the right angle geometrically.

Diagonalize A = [[4,1],[2,3]]. Construct P (eigenvectors) and D (eigenvalues), verify A = PDP⁻¹, and use it to compute Aⁿ quickly.

For A = [[3, 1],[0, 2]], find the characteristic polynomial, its roots, and factor it completely.

Find the eigenvalues and eigenvectors of A = [[4,1],[2,3]]. Show that eigenvectors only scale (they are not rotated) under A.

Build the matrix for a 45° counter-clockwise rotation in the plane. Show how it rotates a sample vector and preserves lengths.

Consider T(x, y) = (2x + y, x − y). Show it is linear, and visualize how it deforms the unit square.

Find the column space of A = [[1,0],[0,1],[1,1]]. Describe which output vectors b make A·x = b solvable.

Find the null space of A = [[1,2,1],[2,4,2]]. Describe all solutions to A·x = 0.

For a 3×4 matrix with rank 2, find the nullity. Explain the theorem and verify with an explicit example.

Find the rank of A = [[1,2,3],[4,5,6],[7,8,9]]. Reduce to echelon form and count pivots.

Find a basis for the column space of A = [[1,2,3],[2,4,5],[3,6,8]]. Show which columns are pivot columns.

Are v₁ = [1,2,3], v₂ = [4,5,6], v₃ = [5,7,9] linearly independent? If not, find an explicit dependence relation.

Express the vector [7, 4] as a linear combination of [1, 2] and [3, -1]. Find the scalar coefficients.

Show that the set of all 2D vectors through the origin forms a vector space. Test closure under addition and scalar multiplication.

Decompose A = [[2,1],[6,4]] as A = LU with L lower triangular (1s on diagonal) and U upper triangular. Verify the product.

Solve the system 3x + 2y = 5, x + 4y = 7 using Cramer's rule. Show the ratios of determinants.