Core Subject Knowledge & Technical Concepts Flashcards

The eigenvalues of the matrix [[2, 1], [0, 3]] are:

Answer: 2 and 3

For an upper triangular matrix, the eigenvalues are the diagonal elements: 2 and 3.

The particular solution of the ODE y'' + 4y = sin(2x) exhibits resonance because:

Answer: The forcing frequency equals a natural frequency of the homogeneous system

Resonance occurs when the forcing frequency (2) equals the natural frequency (√4 = 2) of the homogeneous equation.

The Fourier series of an odd function contains only:

Answer: Sine terms

An odd function f(-x) = -f(x) has zero cosine coefficients (an = 0), so its Fourier series contains only sine terms.

The rank of a matrix equals the number of:

Answer: Linearly independent rows (or columns)

The rank of a matrix is the number of linearly independent rows, which equals the number of linearly independent columns.

Using the trapezoidal rule with n equal intervals of width h, the error is of order:

Answer: O(h²)

The global truncation error of the trapezoidal rule is O(h²), making it a second-order method.

The probability that a standard normal random variable Z satisfies P(Z > 1.96) is approximately:

Answer: 0.025

For a standard normal, P(Z > 1.96) ≈ 0.025, which is why 1.96 defines the 95% confidence interval boundary.

The z-transform of the unit impulse sequence δ[n] is:

Answer: 1

The z-transform of δ[n] = Σδ[n]z⁻ⁿ = z⁰ = 1, valid for all z except z = 0.