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Math Formula Cheat Sheet

"Source" points to the handouts under 学习路线/阶段/.

输入关键词开始查 —— 多个词用空格隔开（都要命中）。

1 Limits

What I want to find Formula Symbol meaning When to use Source Common mistake sympy check
Limit of a function at a point $\lim\limits_{x\to a}f(x)$ $a$ is the point approached find the trend, test continuity Stage 01–02 plugging in directly is not always the answer (it may be $0/0$) limit(sin(x)/x, x, 0) → 1
Limit of a sequence (ε-N definition) $\forall\varepsilon>0,\ \exists N,\ n>N \Rightarrow \lvert a_n-A\rvert<\varepsilon$ ε arbitrarily small, N is "which term it starts from" proving a limit, not computing one Stage 01 dropping "$\forall\varepsilon$" loses the "arbitrarily small" limit(1/n, n, oo) → 0
Limit of a function (ε-δ definition) $\forall\varepsilon>0,\ \exists\delta>0,\ 0<\lvert x-x_0\rvert<\delta \Rightarrow \lvert f(x)-A\rvert<\varepsilon$ δ is the radius of the neighbourhood of x prove the limit exists Stage 02 throwing away $0<\lvert x-x_0\rvert$ — the point itself may be undefined —
$0/0$-type limit use equivalent infinitesimals or L'Hôpital — direct substitution gives $0/0$ Stage 05 if it is not $0/0$ or $\infty/\infty$ you cannot use L'Hôpital limit((exp(x)-1)/x, x, 0) → 1
$\infty/\infty$-type limit L'Hôpital, or compare highest powers — direct substitution gives $\infty/\infty$ Stage 05 L'Hôpital may loop without converging; try something else limit(x**2/exp(x), x, oo) → 0
Left / right limits $\lim_{x\to a^-}f$, $\lim_{x\to a^+}f$ left = from the smaller side, right = from the larger side piecewise functions, testing whether a limit exists Stage 02 left ≠ right ⇒ the limit does not exist limit(1/x, x, 0, '-') → -oo

Core identities (equivalent infinitesimals as $x\to0$):

$$\sin x \sim x, \quad \tan x \sim x, \quad 1-\cos x \sim \frac{x^2}{2}, \quad e^x-1 \sim x, \quad \ln(1+x)\sim x$$

⚠️ "Equivalent infinitesimals" may only be substituted inside multiplication and division, never casually inside addition and subtraction.
Work out the order of the $o(x)$ term before substituting.

2 Derivatives and Differentials

What I want to find Formula Symbol meaning When to use Source Common mistake sympy check
Instantaneous rate of change (definition) $f'(x)=\lim\limits_{\Delta x\to0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}$ Δx is the increment of the independent variable differentiate from the definition, prove differentiability Stage 03 forgetting this is a "limit", thinking you may just set Δx=0 diff(x**2, x) → 2*x
Power function $(x^n)' = nx^{n-1}$ n any real number the most basic rule Stage 04 for $n=0$ it is 0, don't forget diff(x**5, x) → 5*x**4
Product rule $(uv)' = u'v + uv'$ — two functions multiplied Stage 04 not $u'v'$ (the classic mistake) diff(x*sin(x), x) → x*cos(x)+sin(x)
Quotient rule $\left(\dfrac uv\right)' = \dfrac{u'v-uv'}{v^2}$ v ≠ 0 two functions divided Stage 04 the numerator is $u'v - uv'$; reverse the order and it is wrong diff(sin(x)/x, x)
Chain rule ⭐ $[f(g(x))]' = f'(g(x))\cdot g'(x)$ outer derivative × inner derivative any composite function Stage 04 omitting the inner derivative $g'(x)$ diff(sin(x**2), x) → 2*x*cos(x**2)
Trigonometric $(\sin x)'=\cos x$, $(\cos x)'=-\sin x$ — — Stage 04 differentiating $\cos$ carries a minus sign diff(cos(x), x) → -sin(x)
Exponential / logarithm $(e^x)'=e^x$, $(\ln x)'=\dfrac1x$ x > 0 — Stage 04 the derivative of $\ln\lvert x\rvert$ is also $1/x$ diff(log(x), x) → 1/x
Inverse trig $(\arctan x)'=\dfrac{1}{1+x^2}$, $(\arcsin x)'=\dfrac{1}{\sqrt{1-x^2}}$ — you meet these in reverse when integrating Stage 04 don't mix the denominators up with the $\arcsin$ one diff(atan(x), x) → 1/(x**2+1)
Testing monotonicity $f'(x)>0$ ⇒ increasing; $f'(x)<0$ ⇒ decreasing — sketch graphs, find extrema Stage 06 a point with $f'=0$ is not necessarily an extremum (e.g. $x^3$ at 0) solve(diff(x**3-3*x, x), x)
Second derivative for concavity $f''>0$ ⇒ opens upward (concave); $f''<0$ ⇒ opens downward (convex) — determine the direction of bending Stage 07 ⚠️ the Chinese and English conventions for "concave/convex" are opposite — check your textbook first diff(x**2, x, 2) → 2
Finding extrema (two steps) ① solve $f'(x)=0$ for the stationary points ② check the sign of $f''$ — optimisation Stage 06 when $f''=0$ the test tells you nothing; find another way diff(x**3-3*x, x, 2).subs(x, 1) → 6
Tangent line equation $y - f(a) = f'(a)(x-a)$ a is the x-coordinate of the point of tangency find tangent / normal lines Stage 03 forgetting to compute $f(a)$ first diff(x**2,x).subs(x,1) → 2

3 Mean Value Theorem and L'Hôpital

What I want to find Formula Symbol meaning When to use Source Common mistake sympy check
Rolle's theorem $f(a)=f(b)$ ⇒ $\exists\xi,\ f'(\xi)=0$ ξ lies inside (a,b) prove "there is a point where the derivative is 0" Stage 05 three hypotheses (continuous, differentiable, equal endpoint values) — all are required —
Lagrange mean value theorem $\dfrac{f(b)-f(a)}{b-a} = f'(\xi)$ ξ lies inside (a,b) link the "average rate of change" to the "instantaneous rate of change" Stage 05 — —
L'Hôpital's rule $\lim\dfrac{f}{g} = \lim\dfrac{f'}{g'}$ only for $0/0$ or $\infty/\infty$ limits of indeterminate forms Stage 05 ⚠️ using it on a non-indeterminate form ⇒ every answer is wrong limit(sin(x)/x, x, 0)
When L'Hôpital fails when the limit of $f'/g'$ does not exist, the original limit may still exist — when one attempt does not converge Stage 05 don't hammer at it; switch to equivalent infinitesimals or Taylor limit((x+sin(x))/x, x, oo) → 1

4 Indefinite Integrals

What I want to find Formula Symbol meaning When to use Source Common mistake sympy check
Antiderivative / indefinite integral $\int f(x)dx = F(x)+C$, where $F'=f$ C is an arbitrary constant the inverse operation of differentiation Stage 11 ⚠️ forgetting +C integrate(2*x, x) → x**2
Power rule for integrals $\int x^n dx = \dfrac{x^{n+1}}{n+1}+C$ $n \ne -1$ the most used Stage 11 fails when $n=-1$ (division by 0) integrate(x**5, x) → x**6/6
Integral of $1/x$ $\int \dfrac1x dx = \ln\lvert x\rvert+C$ it is the $n=-1$ case — Stage 11 don't drop the absolute value integrate(1/x, x) → log(x)
Exponential / trig $\int e^x dx = e^x+C$
$\int \sin x\,dx = -\cos x+C$		 — — Stage 11 ⚠️ integrating $\sin$ carries a minus sign integrate(sin(x), x) → -cos(x)
Reverse of the inverse-trig derivative $\int \dfrac{1}{1+x^2}dx = \arctan x+C$ — when the denominator is $1+x^2$ Stage 11 don't mix it up with the $\arcsin$ one integrate(1/(1+x**2), x) → atan(x)
Substitution (first kind) $\int f(g(x))g'(x)dx = \int f(u)du$ let $u=g(x)$ the integrand contains "the derivative of some function" Stage 11 forgetting to substitute back to $x$ integrate(2*x*cos(x**2), x) → sin(x**2)
Integration by parts ⭐ $\int u\,dv = uv - \int v\,du$ choose u as the thing that "gets simpler when differentiated" a product of two different types of function Stage 11 the sign is "minus", and the order is easy to reverse integrate(x*exp(x), x) → (x-1)*exp(x)
How to choose $u$ for parts LIATE: Logarithm > Inverse trig > Algebraic > Trig > Exponential (Chinese mnemonic: 反对幂指三) which one to take as $u$ Stage 11 choosing wrong makes it more and more complicated integrate(log(x), x) → x*log(x)-x
The "insert a 1" trick $\int\ln x\,dx = \int \ln x\cdot 1\,dx$ take $dv=dx$, $v=x$ when $\ln x$ or $\arctan x$ appears alone Stage 11 not thinking to "insert a 1" integrate(log(x), x)
Parts twice, "wrapping back" $I = \cdots - I$ ⇒ $2I=\cdots$ treat the original integral as an unknown things like $\int e^x\sin x\,dx$ Stage 11 forgetting that after "wrapping back" you divide by 2 integrate(exp(x)*sin(x), x)
Trigonometric substitution (second kind) $\sqrt{a^2-x^2} \Rightarrow x=a\sin t$ — to remove the square root Stage 11 after substituting you must change the limits integrate(sqrt(1-x**2), x)

5 Definite Integrals and Applications

What I want to find Formula Symbol meaning When to use Source Common mistake sympy check
Definite integral (Riemann sum definition) $\int_a^b f = \lim\limits_{n\to\infty}\sum f(x_i)\Delta x$ Δx = (b−a)/n understand the definition, numerical approximation Stage 12 confusing it with the indefinite integral —
Newton–Leibniz ⭐ $\int_a^b f(x)dx = F(b)-F(a)$ F is any antiderivative everything for evaluating a definite integral rests on it Stage 12 ⚠️ sign errors when plugging in the limits (e.g. $[-\cos x]_0^\pi$) integrate(x**2, (x,0,1)) → 1/3
Interval additivity / swapping limits $\int_a^b f = -\int_b^a f$, $\int_a^b=\int_a^c+\int_c^b$ — integrating piecewise Stage 12 swapping the limits changes the sign —
Odd function on a symmetric interval $\int_{-a}^{a} f = 0$ (f odd) — know it is 0 without computing Stage 12 not spotting the parity, wasting half the work integrate(x**3, (x,-1,1)) → 0
Even function on a symmetric interval $\int_{-a}^{a} f = 2\int_0^a f$ — — Stage 12 — integrate(x**2, (x,-1,1)) → 2/3
Fundamental theorem of calculus $\dfrac{d}{dx}\int_a^x f(t)dt = f(x)$ holds even when the antiderivative has no closed form differentiate a variable-limit integral Stage 12 forgetting that a constant lower limit contributes 0 diff(integrate(exp(-t**2),(t,0,x)), x) → exp(-x**2)
Area between two curves $S=\int_a^b(\text{upper}-\text{lower})dx$ — find the area of a figure Stage 12 getting "which one is on top" wrong ⇒ a negative answer integrate(x-x**2, (x,0,1)) → 1/6
Average value of a function $\bar f = \dfrac{1}{b-a}\int_a^b f\,dx$ — average speed, etc. Stage 12 forgetting to divide by the interval length integrate(x**2, (x,0,3))/3 → 3
Volume of a solid of revolution $V = \pi\int_a^b [f(x)]^2 dx$ rotate about the x-axis find a volume Stage 12 dropping the π or forgetting to square pi*integrate(x**4, (x,0,1)) → pi/5
Convergence test for improper integrals $\int_1^\infty\dfrac{1}{x^p}dx$: $p>1$ converges, $p\le1$ diverges p is the exponent integrals over an infinite interval Stage 12 ⚠️ $\int_1^\infty 1/x$ diverges integrate(1/x, (x,1,oo)) → oo
Numerical integration (when it cannot be done) $\int_a^b f \approx$ numerical methods — no elementary antiderivative Stage 12 — float(integrate(exp(-x**2),(x,0,1)))

6 Determinants

What I want to find Formula Symbol meaning When to use Source Common mistake sympy check
2×2 determinant $\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc$ — solve a 2-variable system Stage 08 forgetting to subtract the anti-diagonal term Matrix([[2,1],[1,3]]).det() → 5
3×3 (Sarrus' rule) add the 3 main-direction terms − add the 3 anti-direction terms — valid only for 2×2 and 3×3 Stage 08 ⚠️ cannot be used from 4×4 up! also easy to miss one diagonal Matrix([[1,2,3],[4,5,6],[7,8,10]]).det() → -3
Conditions for determinant = 0 ① two rows equal ② one row is a multiple of another ③ some row is all 0 — know it is 0 without computing Stage 08 not spotting it and grinding through the computation Matrix([[1,2],[2,4]]).det() → 0
Determinant property (row swap flips sign) swapping two rows ⇒ $\det$ changes sign — simplify a computation Stage 08 — —
Determinant property (adding a multiple of a row) $R_i + kR_j$ ⇒ $\det$ unchanged — reduce to upper triangular (what a computer does) Stage 08 confusing it with "multiply a row by k" (that one multiplies by k) —
Triangular determinant upper/lower triangular ⟹ $\det$ = product of the diagonal — one step to the answer once triangular Stage 08 — Matrix([[1,2,3],[0,4,5],[0,0,6]]).det() → 24
Determinant of a product $\det(AB)=\det(A)\det(B)$ — simplify a computation Stage 08 — (Matrix([[1,2],[3,4]])*Matrix([[0,1],[1,0]])).det()
Transpose invariance $\det(A^T)=\det(A)$ — — Stage 08 — —
Cramer's rule $x_i = D_i / D$, $D_i$ is what you get by replacing column i with the constant column requires $D\ne0$ solve n equations in n unknowns (theoretical use) Stage 08 ⚠️ not used in actual computation ($O(n\cdot n!)$ is too slow) Matrix([[1,1,1],[2,-1,1],[1,2,-1]]).det() → 7

7 Matrix Operations

What I want to find Formula Symbol meaning When to use Source Common mistake sympy check
Matrix multiplication $(AB)_{ij}=\sum_k a_{ik}b_{kj}$ (left row × right column) dimensions must be $(m\times n)(n\times p)$ composing transformations Stage 09 ⚠️ the left column count must equal the right row count Matrix([[1,2],[3,4]])*Matrix([[0,1],[1,0]])
Multiplication is not commutative $AB \ne BA$ (in general) — at all times Stage 09 ⚠️ the classic mistake: assuming you may swap Matrix([[1,2],[3,4]])*Matrix([[0,1],[1,0]]) == Matrix([[0,1],[1,0]])*Matrix([[1,2],[3,4]]) → False
Cancellation fails $AB=AC \not\Rightarrow B=C$ — matrix equations Stage 09 also "assuming you can cancel as with numbers" —
Zero divisors $AB=O$ but $A\ne O,B\ne O$ — — Stage 09 for numbers "a product of 0 means one factor is 0"; for matrices it does not hold —
Transpose $(A^T)_{ij}=a_{ji}$ rows and columns swap — Stage 09 — Matrix([[1,2,3],[4,5,6]]).T
Transpose of a product $(AB)^T = B^TA^T$ — — Stage 09 ⚠️⚠️ reverse the order (A*B).T == B.T*A.T → True
Identity matrix $AI = IA = A$ all 1s on the diagonal the "1" of matrices Stage 09 — Matrix([[1,2],[3,4]])*eye(2)
Trace $\text{tr}(A)=\sum_i a_{ii}$ (sum of the main diagonal) — equals the sum of the eigenvalues Stage 13 — Matrix([[2,1],[1,2]]).trace() → 4
Rank $\text{rank}(A)$ = number of independent rows/columns $\le\min(m,n)$ determine the number of solutions Stage 10 the rank cannot exceed the smaller of the two dimensions Matrix([[1,2,3],[2,4,6],[1,1,1]]).rank() → 2
Reduced row echelon form (rref) use row operations to make each column's pivot 1 and the rest 0 — the standard way to solve systems / find the rank Stage 10 — Matrix([[1,2],[3,4]]).rref()

8 Inverse Matrices

What I want to find Formula Symbol meaning When to use Source Common mistake sympy check
Definition of the inverse $AA^{-1} = A^{-1}A = I$ must be square with $\det\ne0$ "division for matrices" Stage 09 a non-square matrix has no inverse; $\det=0$ means no inverse —
2×2 inverse formula ⭐ $\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1}=\dfrac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$ — by hand for order 2 Stage 09 ⚠️ forgetting to divide by the determinant Matrix([[1,2],[3,4]]).inv() → [[-2,1],[3/2,-1/2]]
Mnemonic swap the main diagonal, flip the sign of the anti-diagonal, divide by the determinant — memorise it Stage 09 forgetting to flip the sign on the anti-diagonal —
Adjugate method (order 3) $A^{-1} = \dfrac{1}{\det A}\operatorname{adj}(A)$ adj = the transpose of the cofactor matrix by hand for order 3 (very tedious) Stage 09 forgetting the transpose step Matrix([[1,2,3],[0,1,4],[5,6,0]]).adjugate()
Inverse of a product $(AB)^{-1} = B^{-1}A^{-1}$ — — Stage 09 ⚠️⚠️ reverse the order (the same rule as the transpose) (A*B).inv() == B.inv()*A.inv() → True
Inverse of a transpose $(A^T)^{-1} = (A^{-1})^T$ — the order may be swapped Stage 09 — (A.T).inv() == A.inv().T → True
Determinant of an inverse $\det(A^{-1}) = 1/\det(A)$ — — Stage 09 — Matrix([[1,2],[3,4]]).inv().det() → -1/2
Solving a system with the inverse $AX=b \Rightarrow X=A^{-1}b$ — when solving for several b with the same A Stage 09 for a single solve, solve is faster Matrix([[2,1],[1,-3]]).inv()*Matrix([5,-1])

⚠️ In practice use np.linalg.solve(A, b) — do not invert first and then multiply.
It is faster and numerically more stable.

9 Systems of Linear Equations

What I want to find Formula Symbol meaning When to use Source Common mistake sympy check
Decision table for solutions ⭐ $r_A < r_b$ ⇒ no solution
$r_A = r_b = n$ ⇒ unique solution
$r_A = r_b < n$ ⇒ infinitely many solutions		 $r_A=\text{rank}(A)$, $r_b=\text{rank}([A\mid b])$, $n$=number of unknowns decide without solving the system Stage 10 ⚠️ computing only $\text{rank}(A)$ is not enough; you must also compute the rank of the augmented matrix see below
Homogeneous system has a nonzero solution $\text{rank}(A) < n$ — test "is there a nonzero solution" Stage 10 a homogeneous system always has the zero solution; the question is whether there is a nonzero one Matrix([[1,2,3],[2,4,6],[1,1,1]]).rank() < 3 → True
General solution of a homogeneous system $X = k_1\xi_1 + k_2\xi_2 + \cdots$ fundamental system of solutions $\xi_i$ write out all solutions Stage 10 number of free variables = $n-\text{rank}$ Matrix([[1,2,3],[2,4,6],[1,1,1]]).nullspace()
General solution of a non-homogeneous system $X = \eta^* + k_1\xi_1+\cdots$ η is one particular solution* write out all solutions Stage 10 forgetting to add the homogeneous part linsolve((A, b), (x,y,z))
Linear dependence / independence if some $k_i$, not all zero, satisfy $\sum k_i\alpha_i=0$ ⇒ dependent — test "are there redundant vectors" Stage 10 — Matrix.hstack(v1,v2,v3).rank() == 3 → independent
More than $n$ vectors in $n$ dimensions always linearly dependent — a quick test Stage 10 forgetting this wastes the work —

Terminal check for the decision table:

# example with no solution (rank(A)=1, rank([A|b])=2)
~/miniconda3/bin/python -c "
from sympy import Matrix
A=Matrix([[1,2],[2,4]]); b=Matrix([1,3])
print('rank(A)=',A.rank(),' rank([A|b])=',A.row_join(b).rank(),' -> not equal => no solution')"

# example with infinitely many solutions (the two ranks are equal and < number of unknowns)
~/miniconda3/bin/python -c "
from sympy import Matrix
A=Matrix([[1,2,3],[2,4,6]]); b=Matrix([1,2])
print('rank(A)=',A.rank(),' rank([A|b])=',A.row_join(b).rank(),' unknowns=3 -> infinitely many solutions')"

10 Eigenvalues and Diagonalisation

What I want to find Formula Symbol meaning When to use Source Common mistake sympy check
Eigenvalues and eigenvectors $Av=\lambda v$, $v\ne0$ λ eigenvalue, v eigenvector find the "direction that does not turn" Stage 13 an eigenvector cannot be the zero vector Matrix([[2,1],[1,2]]).eigenvects()
Characteristic equation ⭐ $\det(A-\lambda I)=0$ you must subtract λI find the eigenvalues Stage 13 ⚠️ writing $\det(A-\lambda)$ is completely wrong Matrix([[2,1],[1,2]]).charpoly(lam)
Sum of the eigenvalues $\sum_i\lambda_i = \text{tr}(A)$ trace = sum of the main diagonal a one-second check Stage 13 use this to spot an arithmetic error immediately sum(Matrix([[2,1],[1,2]]).eigenvals()) → 4
Product of the eigenvalues $\prod_i\lambda_i = \det(A)$ — check the result Stage 13 — Matrix([[2,1],[1,2]]).det() → 3
Eigenvalues of a triangular matrix they are the diagonal entries — the fastest case Stage 13 — Matrix([[1,2,3],[0,4,5],[0,0,6]]).eigenvals()
Eigenvalues of a symmetric matrix always real, with eigenvectors mutually orthogonal — covariance matrices, etc. Stage 13 — Matrix([[2,1],[1,2]]).eigenvals() → {1:1, 3:1}
Diagonalisation $A = P\Lambda P^{-1}$, or $\Lambda = P^{-1}AP$ columns of P = eigenvectors, Λ = diagonal of eigenvalues make a matrix simpler Stage 15 ⚠️ the column order of P must match the diagonal order of Λ Matrix([[2,1],[1,2]]).diagonalize()
Condition for diagonalisability A has $n$ linearly independent eigenvectors — decide whether it can be diagonalised Stage 15 with a repeated root it may still be diagonalisable Matrix([[2,1],[0,2]]).is_diagonalizable() → False
Matrix power (via diagonalisation) $A^k = P\Lambda^k P^{-1}$ Λ^k just raises the diagonal entries to the power compute a high power Stage 15 multiplying out directly is slow (Matrix([[1,1],[1,0]])**10) → Fibonacci
Orthogonal diagonalisation of a real symmetric matrix $A = Q\Lambda Q^T$, $Q^{-1}=Q^T$ Q is an orthogonal matrix PCA, covariance matrices Stage 15 only symmetric matrices allow this (for a general matrix $P^{-1}\ne P^T$) Matrix([[2,1],[1,2]]).diagonalize(normalize=True)
Geometric meaning of diagonalisation switch to the basis of eigenvectors and the transformation becomes independent stretching along each direction — build intuition Stage 15 — —
Long-term behaviour $\|\lambda_{\max}\|>1$ diverges; $=1$ stable; $<1$ decays — analysing dynamical systems Stage 15 — —

11 SVD

What I want to find Formula Symbol meaning When to use Source Common mistake sympy check
Singular value decomposition ⭐ $A = U\Sigma V^T$ U ($m\times m$) orthogonal, Σ ($m\times n$) diagonal non-negative, $V^T$ ($n\times n$) orthogonal every matrix has one Stage 16 ⚠️ U and V have different sizes; truncate when reconstructing U,S,Vt = np.linalg.svd(A)
Geometric meaning rotate → stretch along the coordinate axes → rotate — build intuition Stage 16 — —
Relation between singular values and eigenvalues the eigenvalues of $A^TA$ $= \sigma_i^2$ σ is a singular value understand where the SVD comes from Stage 16 in practice you do not compute it this way (poor accuracy) np.linalg.eigvalsh(A.T@A)
Symmetric matrix $\sigma_i = \|\lambda_i\|$ — the special case of a symmetric matrix Stage 16 ⚠️ eigenvalues may be negative; singular values are never negative np.linalg.svd(C)[1]
Low-rank approximation (Eckart–Young) $A\approx U_k\Sigma_kV_k^T$ (keep only the first k singular values) — image compression, dimensionality reduction, denoising Stage 16 what is truncated is the "least important" directions see below
Pseudoinverse $A^+ = V\Sigma^+U^T$ Σ⁺ = take the reciprocal of the nonzero singular values, then transpose the fallback when $A$ is not invertible (regression) Stage 16 — np.linalg.pinv(A)
The other face of PCA SVD of centred data ⇒ right singular vectors = principal component directions singular value² ÷(n−1) = variance of each principal component dimensionality reduction Stage 16 — —

A hands-on test of low-rank approximation (the example from the Stage 16 handout):

~/miniconda3/bin/python -c "
import numpy as np
np.random.seed(0)
X = np.outer(np.arange(1,11), np.arange(1,6)) + np.random.normal(0,0.5,(10,5))
U,S,Vt = np.linalg.svd(X)
for k in [1,2,3]:
    Xk = U[:,:k] @ np.diag(S[:k]) @ Vt[:k]
    print(f'first {k}: relative error {np.linalg.norm(X-Xk)/np.linalg.norm(X):.4f}')"
# first 1: relative error 0.0222   ← 16 numbers reconstruct 50 numbers, error 2%
# first 2: relative error 0.0166
# first 3: relative error 0.0103

12 Common Constants and Equivalent Infinitesimals

Name Value / Formula sympy syntax
Pi π $3.14159265\ldots$ sp.pi
Euler's number e $2.71828182\ldots$ sp.E
Infinity — sp.oo
Golden ratio φ $\dfrac{1+\sqrt5}{2}=1.61803398\ldots$ (1+sp.sqrt(5))/2
Imaginary unit i $\sqrt{-1}$ sp.I

Equivalent infinitesimals ($x\to0$):

Expression Equivalent to sympy check
$\sin x$ $x$ limit(sin(x)/x, x, 0) → 1
$\tan x$ $x$ limit(tan(x)/x, x, 0) → 1
$1-\cos x$ $\dfrac{x^2}{2}$ limit((1-cos(x))/x**2, x, 0) → 1/2
$e^x-1$ $x$ limit((exp(x)-1)/x, x, 0) → 1
$\ln(1+x)$ $x$ limit(log(1+x)/x, x, 0) → 1
$(1+x)^\alpha-1$ $\alpha x$ limit(((1+x)**0.5-1)/x, x, 0) → 0.5

13 High-Frequency Mistakes (must-read before the exam)

These are counted from Stages 01–16, ordered by how often they occur.

# Mistake Correct
1 $\int\sin x\,dx = \cos x$ it is $-\cos x+C$
2 forgetting $+C$ on an indefinite integral write it every time
3 $(AB)^{-1} = A^{-1}B^{-1}$ it is $B^{-1}A^{-1}$, the order is reversed
4 $(AB)^T = A^TB^T$ it is $B^TA^T$, the order is reversed
5 forgetting to change the limits after substitution once you change to $u$, change the limits
6 computing $[-\cos x]_0^\pi$ as 0 $-\cos\pi=+1$, the answer is 2
7 forgetting to write $I$ in $\det(A-\lambda I)$ subtract from the main diagonal only
8 giving only eigenvalues, not eigenvectors the two must be paired
9 forgetting the $D\ne0$ precondition of Cramer's rule compute $D$ first
10 computing only $\text{rank}(A)$, not $\text{rank}([A\mid b])$ compute both
11 using L'Hôpital on a non-indeterminate form first confirm it is $0/0$ or $\infty/\infty$
12 using == to compare floats use math.isclose

⭐ Items 3 and 4 are the sneakiest — they go wrong exactly when they look right.
Mnemonic: both the inverse and the transpose "turn the order around", like putting socks and shoes on and off.

14 How to use this sheet

Step What to do
1 search here when you get stuck on a problem (Cmd+F the "What I want to find" column)
2 read "Formula" and "When to use"
3 when unsure, copy the "sympy check" line into a terminal and run it
4 read the "Common mistake" column — those are the pits others fell into, and the ones you are about to

Template command for running sympy:

~/miniconda3/bin/python -c "
import sympy as sp
x, y, z = sp.symbols('x y z')
print(sp.diff(x**2, x))
print(sp.integrate(x**2, x))
print(sp.limit(sp.sin(x)/x, x, 0))
"

💡 Save the few lines you use most as a shortcut, for example in .zshrc:

alias math='~/miniconda3/bin/python -c'

Then you can just run math "import sympy as s;x=s.symbols('x');print(s.diff(x**2,x))".

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Last updated: 2026-09-26