Linear Algebra and Calculus

Derivatives and limits

The derivative of a function at a point gives the slope of the tangent line to its graph there, so the coordinates of the point of tangency only locate the contact and do not supply the slope. For a power function the power rule applies, \frac{d}{dx}x^{n}=nx^{n-1}. For example, the slope of y=x^{2} at x=3 is 2\times 3=6.

Limits of the form \frac{0}{0} are evaluated with standard results such as \lim_{x\to 0}\frac{\sin x}{x}=1 and \lim_{x\to 0}\frac{\ln(1+x)}{x}=1. A factor that vanishes in the numerator can decide the answer, since \frac{x^{2}}{\sin x}=x\cdot\frac{x}{\sin x} tends to 0\cdot 1=0. A constant in the denominator simply divides the limit, so \lim_{x\to 0}\frac{\ln(1+x)}{2\sin x}=\frac{1}{2}.

When square roots produce an indeterminate form, multiply the numerator and denominator by the conjugate. Here \frac{\sqrt{1+x}-1}{x}=\frac{1}{\sqrt{1+x}+1}, which tends to \frac{1}{2} as x\to 0.

For extreme values on a closed interval, compare the endpoint values with the values at critical points where f'(x)=0 or f'(x) is undefined. The second derivative test classifies a critical point c: f''(c)<0 gives a local maximum and f''(c)>0 gives a local minimum. For a polynomial of even degree with positive leading coefficient, f(x)\to\infty as |x|\to\infty, so the smallest critical value is the global minimum.

Matrix multiplication and inverse

Matrix multiplication is associative but not commutative, and the reversal rules for transposes and inverses are frequently tested. For conformable matrices, the transpose of a product reverses the order of the factors, (\mathbf{PQ})^{T}=\mathbf{Q}^{T}\mathbf{P}^{T}, and this holds whatever the shapes of the factors. For invertible square matrices the same reversal applies to inverses, (\mathbf{RS})^{-1}=\mathbf{S}^{-1}\mathbf{R}^{-1}, because (\mathbf{RS})(\mathbf{S}^{-1}\mathbf{R}^{-1})=\mathbf{I}. Inverting twice returns the original matrix, (\mathbf{P}^{-1})^{-1}=\mathbf{P}.

A square matrix is symmetric when \mathbf{A}^{T}=\mathbf{A} and skew-symmetric when \mathbf{A}^{T}=-\mathbf{A}. A relation of the form \mathbf{Q}=-\mathbf{Q}^{T} therefore defines a skew-symmetric matrix and must not be confused with symmetry.

Determinants are multiplicative, \det(\mathbf{PQ})=\det\mathbf{P}\,\det\mathbf{Q}, and transposition does not change the determinant, \det\mathbf{A}^{T}=\det\mathbf{A}. For a 2\times 2 matrix, \det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc. For example, if \det\mathbf{P}=3 and \det\mathbf{Q}=4, then \det(\mathbf{QP})=12.

Two relations involving \mathbf{P}+\mathbf{Q} and \mathbf{P}-\mathbf{Q} can be combined to isolate one matrix, because (\mathbf{P}+\mathbf{Q})-(\mathbf{P}-\mathbf{Q})=2\mathbf{Q}. Dividing the difference by 2 then gives \mathbf{Q}.

A linear matrix equation such as 2\mathbf{X}+\mathbf{A}=\mathbf{B} is solved like a scalar equation: \mathbf{X}=\frac{1}{2}(\mathbf{B}-\mathbf{A}), with subtraction and scalar division performed entry by entry.

Definite integrals and area under curves

A definite integral \int_{a}^{b}f(x)\,dx gives the signed area between the graph of f and the x-axis, which equals the geometric area when f\ge 0 on [a,b]. For instance, \int_{0}^{1}x^{3}\,dx=\frac{1}{4} is the area under y=x^{3} from x=0 to x=1.

For a region between two curves on [a,b], the area is \int_{a}^{b}(f_{\text{upper}}-f_{\text{lower}})\,dx, and the upper curve must be identified by comparing values across the whole interval. For example, between y=x and y=x^{2} on [0,1] the line is the upper curve, and the area is \int_{0}^{1}(x-x^{2})\,dx=\frac{1}{6}.

Several standard results recur. \int_{0}^{\pi/2}\sin^{2}x\,dx=\frac{\pi}{4} follows from \sin^{2}x=\frac{1-\cos 2x}{2}. The Dirichlet integral \int_{0}^{\infty}\frac{\sin ax}{x}\,dx=\frac{\pi}{2} holds for every a>0, so the value does not depend on the positive constant a, and only a constant multiplier outside the integral needs to be tracked.

A region bounded by a straight line in the first quadrant and the coordinate axes is a right triangle, so its area is half the product of the intercepts. For the line y=c-x with c>0, the area is \frac{c^{2}}{2}.

Eigenvalues and eigenvectors

An eigenvalue \lambda of a square matrix \mathbf{A} is a scalar for which \det(\mathbf{A}-\lambda\mathbf{I})=0, and expanding this determinant gives the characteristic equation. For a 2\times 2 matrix the characteristic equation is \lambda^{2}-(\text{trace})\,\lambda+\det\mathbf{A}=0. For example, \begin{bmatrix}1&2\\3&4\end{bmatrix} has trace 5 and determinant -2, so its characteristic equation is \lambda^{2}-5\lambda-2=0.

For a symmetric 2\times 2 matrix of the form \begin{bmatrix}a&b\\b&a\end{bmatrix}, the eigenvalues are a+b and a-b. For instance, \begin{bmatrix}5&2\\2&5\end{bmatrix} has eigenvalues 7 and 3. A candidate value in a multiple-choice list can be verified by substituting it into \det(\mathbf{A}-\lambda\mathbf{I}) and checking whether the result is zero.

For an n\times n matrix, the sum of the eigenvalues equals the trace and their product equals the determinant. These two checks quickly eliminate wrong options in a 3\times 3 problem, because a candidate list whose sum differs from the trace cannot be correct.