BMath Bachelor of Mathematics Linear Algebra and Matrix Theory 3

Question 1

What is the inner product (dot product) of vectors u = (1, 2, 3) and v = (4, 5, 6)?

Accepted Answer

32

Answer

u · v = 1×4 + 2×5 + 3×6 = 4 + 10 + 18 = 32.

Question 2

The Gram-Schmidt process is used to:

Accepted Answer

Convert a basis into an orthonormal basis

Answer

Gram-Schmidt orthogonalizes a set of linearly independent vectors into a mutually orthogonal (and normalizable) set.

Question 3

For an n×n matrix, the characteristic polynomial is of degree:

Accepted Answer

n

Answer

The characteristic polynomial det(A − λI) expands to a degree-n polynomial in λ, yielding n eigenvalues (counting multiplicity).

Question 4

Cramer's Rule provides a formula for solving Ax = b using:

Accepted Answer

Ratios of determinants

Answer

Cramer's Rule expresses each variable xᵢ as det(Aᵢ)/det(A), where Aᵢ replaces the i-th column of A with b.

Question 5

A matrix A is diagonalizable if and only if:

Accepted Answer

It has n linearly independent eigenvectors

Answer

An n×n matrix is diagonalizable exactly when it has n linearly independent eigenvectors, allowing it to be written as A = PDP⁻¹.

BMath Bachelor of Mathematics Practice Test