checklist

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📚 Math 2109: The Ultimate Master Study Checklist

👨‍🏫 Instructor 1: Matrices & Laplace Transform

Chapter 1: Matrix Basics, Definitions, and Properties

• Describe the real-world applications of matrices (RGB coloring, DSP, image processing, linear equations).
• Define and identify basic matrix types: Row, Column, Rectangular, Square, Scalar, Diagonal, Identity, and Null matrices. [PYQ: 2017]
• Define, identify, and construct Symmetric and Skew-Symmetric matrices (including proving diagonal elements of skew-symmetric are zero). [PYQ: 2017, 2019, 2024] [Heavily Tested]
• Mathematically prove that every square matrix can be uniquely expressed as the sum of a symmetric and skew-symmetric matrix. [PYQ: 2018, 2019]
• Define and identify Upper and Lower Triangular matrices. [PYQ: 2015]
• Define and identify Hermitian and Skew-Hermitian matrices over the complex field. [PYQ: 2015, 2016, 2017, 2018, 2021, 2025] [Heavily Tested]
• Mathematically prove that every square matrix can be expressed as the sum of a Hermitian and Skew-Hermitian matrix. [PYQ: 2018]
• Define Submatrices and Elementary matrices. [PYQ: 2016, 2023]
• Define and identify Singular, Orthogonal, Idempotent, Nilpotent (and state its order), Periodic, Involutory, and Unitary matrices. [PYQ: 2015, 2017, 2018, 2020, 2023]
• Perform Matrix Block Multiplication by partitioning.
• Prove specific matrix relation properties (e.g., commutation, anti-commutation $(A+B{)}^2=A^2+B^2$, periodicity). [PYQ: 2016, 2018]
• Explain the relationship between the trace of a matrix $tr(A)$ and the trace of its inverse $tr(A^{-1})$ for an invertible matrix. [PYQ: 2023]
• Determine the conditions or specific unknown constants (e.g., finding $c$) under which a given matrix is NOT invertible. [PYQ: 2023]
• Use the Cayley-Hamilton theorem to verify the theorem, compute the inverse of a matrix ($A^{-1}$), and evaluate high-degree matrix polynomials. [PYQ: 2016]

Chapter 2: Rank, Inverse, Canonical Forms & Vector Spaces

• Execute Elementary Row Operations (Transformations) accurately.
• Define the Rank of a matrix. [PYQ: 2019, 2024, 2025]
• Reduce a given matrix to Row-Echelon form to determine its rank. [PYQ: 2015, 2016, 2021, 2022, 2023, 2025] [Heavily Tested]
• Reduce a matrix to its Canonical and Normal forms. [PYQ: 2015, 2017, 2018, 2021]
• Test for invertibility and find the Inverse of a matrix using elementary row transformations (row canonical form). [PYQ: 2015, 2016, 2017, 2018, 2022, 2024, 2025] [Heavily Tested]
• Define a Vector Space and mathematically prove whether a given set of vectors forms a valid vector space. [PYQ: 2019]
• Express a given vector as a linear combination of other given coordinate vectors. [PYQ: 2015]

Chapter 3: System of Linear Equations

• Define homogeneous and non-homogeneous systems of linear equations. [PYQ: 2025]
• Solve systems of equations using Gaussian elimination / the matrix method. [PYQ: 2015, 2022]
• Apply the rank test to evaluate the consistency of a non-homogeneous system of linear equations. [PYQ: 2017, 2019]
• Analyze parametric conditions (finding $\lambda$, $\mu$, $k$) to determine if a system has no solution, a unique solution, or many/infinite solutions. [PYQ: 2020, 2021, 2024, 2025] [Heavily Tested]
• Interpret the geometrical meaning of solutions to a linear system. [PYQ: 2023, 2025]

Chapter 4: Eigenvalues, Eigenvectors & Diagonalization

• Define Eigenvalue, Eigenvector, and Characteristic Equation. [PYQ: 2021, 2023, 2025]
• Calculate eigenvalues and the corresponding eigenvectors (specifically for the largest eigenvalue). [PYQ: 2015, 2017, 2018, 2019, 2021, 2025] [Heavily Tested]
• Evaluate the minimal polynomial for a given matrix. [PYQ: 2025]
• Determine if a matrix is diagonalizable and find the invertible matrix $P$ such that $P^{-1}AP$ is a diagonal matrix. [PYQ: 2018, 2020, 2023, 2024, 2025] [Heavily Tested]
• Find the Spectral and Modal matrices and use them to calculate high matrix powers (e.g., $B^{100}$). [PYQ: 2022, 2023]

Chapter 5: Engineering Applications of Matrices

• Construct a system of linear equations from a given electrical network / circuit diagram and solve for loop/mesh currents ($i_1,i_2,i_3$). [PYQ: 2016, 2019, 2021, 2022, 2023, 2024] [Heavily Tested]
• Set up and solve a linear system to find unknown flow rates in network branches. [PYQ: 2020]
• Formulate a linear system to find a polynomial curve passing through specific coordinate points. [PYQ: 2023]
• Calculate the steady-state temperature distribution at nodes on a metal plate. [PYQ: 2018]

Chapter 6: Laplace Transform

• Define the Laplace Transform and state the sufficient conditions for its existence. [PYQ: 2022, 2024]
• State and apply the First Shifting, Second Shifting, and Convolution theorems. [PYQ: 2022, 2025]
• Compute the Laplace Transform of elementary functions using integration definitions. [PYQ: 2020]
• Compute the Laplace Transform of the Error Function (e.g., $e^t\cdot t\cdot \text{erf}(\sqrt{t})$). [PYQ: 2022]
• Find the Laplace Transform of Periodic Functions (defined graphically or piecewise). [PYQ: 2021]
• Express piecewise functions in terms of the Unit Step Function and evaluate their Laplace transforms. [PYQ: 2021]
• Evaluate improper definite integrals (e.g., ${\int }_0^{\infty }t{e}^{-2t}\cos t,dt$) by applying Laplace transform definitions and properties rather than standard integration. [PYQ: 2016, 2019]

Chapter 7: Inverse Laplace Transform & Applications

• Define the Inverse Laplace transformation. [PYQ: 2025]
• Evaluate Inverse Laplace Transforms using standard properties and Convolution. [PYQ: 2021, 2022, 2024, 2025]
• Solve Ordinary Differential Equations / Initial Value Problems (IVPs) using the Laplace transform method. [PYQ: 2021, 2022, 2023, 2024, 2025] [Heavily Tested]
• Determine the current $i(t)$ or charge $q(t)$ in single-loop circuits (LR, RC, RLC series) given a specific electromotive force $E(t)$ using Laplace transforms. [PYQ: 2021, 2022, 2025]

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👨‍🏫 Instructor 2: Vector Calculus & Fourier Series

Chapter 8: Vector Differentiation & Properties

• Define Scalar point functions and Vector point functions with examples. [PYQ: 2021]
• Define Gradient, Divergence, and Curl and state their physical significance/outcomes. [PYQ: 2021, 2024, 2025]
• Define a Conservative Vector Field. Prove a given vector field is conservative and determine its Scalar Potential ($\varphi$). [PYQ: 2021, 2025]
• Define a Solenoidal Vector. Find constants to ensure a vector field is solenoidal (where Divergence $\nabla \cdot \bar{A}=0$). [PYQ: 2019, 2025]
• Mathematically evaluate complex derivative operator expressions (e.g., $\nabla \cdot [\bar{r}\nabla (\frac{1}{r^3})]$). [PYQ: 2024]
• Prove Laplacian operator (${\nabla }^2$) identities for scalar functions of $r$ (e.g., mathematically prove ${\nabla }^{2}f(r)=f^{\prime \prime }(r)+\frac{2}{r}{f}^{\prime }(r)$ or ${\nabla }^2(\frac{1}{r})=0$). [PYQ: 2019]

Chapter 9: Geometry of Vectors

• Calculate the Directional Derivative of a scalar field in a specified direction. [PYQ: 2021, 2024]
• Calculate the Maximum Directional Derivative (or maximum rate of change of heat/temperature) of a scalar field and determine the specific direction in which it occurs. [PYQ: 2015, 2021]
• Prove that $\text{grad}(\varphi )$ is a vector perpendicular to the surface $\varphi (x,y,z)=c$. [PYQ: 2025]
• Determine the equation for the Tangent Plane and parametric equation for the Normal Line to a given surface at a specific point. [PYQ: 2021, 2024, 2025] [Heavily Tested]
• Find specific constants to make two surfaces orthogonal at a given point. [PYQ: 2025]
• Calculate the angle between two surfaces at a specified intersection point. [PYQ: 2021]
• For a moving particle in a force field, calculate directional velocity, acceleration, curvature, tangential, and normal components at a given time $t$. [PYQ: 2021]

Chapter 10: Vector Integration & Theorems

• Write the mathematical statements of Gauss’s Divergence theorem, Stoke’s theorem, and Green’s theorem. [PYQ: 2025]
• Evaluate the Line Integral ($\int F\cdot dr$) along specific parametric paths or straight-line segments. [PYQ: 2024, 2025]
• Calculate the Work Done moving a particle in a force field along a specific curve or line segment. [PYQ: 2021, 2024, 2025]
• Evaluate Surface Integrals ($\iint F\cdot \hat{n}ds$) using the law of projection over bounded regions. [PYQ: 2024]
• Evaluate Volume Integrals ($\iiint Vdv$) bounded by planes/cylinders. [PYQ: 2025]
• Apply Green’s Theorem in the plane to evaluate integrals for a closed region. [PYQ: 2024]
• Apply Stoke’s Theorem to evaluate the work done / flux over a specified boundary. [PYQ: 2021, 2025]
• Apply Gauss’s Divergence Theorem to find the total flux/surface integral over an entire bounding surface. [PYQ: 2021, 2024, 2025] [Heavily Tested]

Chapter 11: Fourier Series

• Write down Dirichlet’s conditions for the existence of a Fourier Series. [PYQ: 2021, 2024, 2025]
• Find the Fourier series expansion of standard and piecewise functions (e.g., $f(x)=x\cos x$, $f(x)=x$, etc.) in intervals like $(-\pi \le x\le \pi )$ or $(-2<x<2)$. [PYQ: 2021, 2024, 2025] [Heavily Tested]
• Expand functions into Half-Range Fourier Sine Series or Cosine Series. [PYQ: 2019, 2024]
• Formulate the Complex form of the Fourier series for a given function (e.g., $g(t)=e^{-t}$). [PYQ: 2017]
• Deduce infinite numerical series sums (e.g., evaluate ${\sum }_{n=1}^{\infty }\frac{1}{(2n-1{)}^2}$) using the derived Fourier series. [PYQ: 2024, 2025]
• Perform Harmonic Analysis: Given a table of values for a periodic current/function over a period, compute the direct current part and the amplitude of the specific (e.g., second, third) harmonic. [PYQ: 2019]

Chapter 12: Fourier Integral & Transform

• Derive and represent functions using the Fourier Integral. [PYQ: 2021, 2024, 2025]
• Compute the Fourier Transform of piecewise continuous functions (e.g., $f(x)=1-x^2$ for $|x|<1$). [PYQ: 2024]
• Compute the Finite Fourier Sine / Cosine Transform for a given function over a restricted finite interval (e.g., $0<x<4$). [PYQ: 2017]
• Compute the infinite Fourier Cosine Transform and Fourier Sine Transform of given functions (e.g., $f(x)=\frac{1}{1+x^2}$). [PYQ: 2025]
• Obtain the Inverse Fourier Transform for specified frequency domain signals (e.g., $X(j\omega )=\frac{3j\omega +10}{(j\omega {)}^2+6j\omega +8}$). [PYQ: 2025]
• Utilize Fourier integrals to deduce the value of specific improper integrals (e.g., finding the value of ${\int }_0^{\infty }\frac{\sin p}{p}dp$). [PYQ: 2021]
• Solve partial differential equations (like $\frac{\partial u}{\partial t}=\frac{{\partial }^{2}u}{\partial x^2}$) subject to given boundary/initial conditions using Fourier transforms. [PYQ: 2025]

missed above:

Here are the final missing granular tasks you should add to ensure 100% bulletproof coverage:

👨‍🏫 Instructor 1: Matrices & Laplace Transform

Chapter 1 & 2: Matrix Basics, Rank & Algebra

• Calculate the maximum and minimum possible rank for a given non-square matrix (e.g., a $4\times 3$ matrix). [PYQ: 2022]
• Solve linear matrix algebra equations to find unknown matrices (e.g., given $3A-2B=X$ and $-4A+B=Y$, find matrix $B$). [PYQ: 2017]
• Construct a specific matrix $B$ such that $AB$ is a zero matrix, where $B$ must have distinct non-zero columns. [PYQ: 2018]

Chapter 4: Eigenvalues & Eigenvectors

• Verify if specifically given column vectors (e.g., $X_1,X_2$) are valid eigenvectors of a given matrix before diagonalizing. [PYQ: 2018, 2021]

Chapter 6 & 7: Laplace Transform & Applications

• Apply the “Division by $t$” property to find the Laplace transform of functions like $\frac{1-e^{2t}}{t}$ or $\frac{e^{-at}-e^{-bt}}{t}$. [PYQ: 2017, 2020]
• Apply the “Multiplication by $t^n$” property in combination with the First Shifting theorem (e.g., find LT of $t\cos (2t)e^{-2t}$). [PYQ: 2017]
• Find the original function $Lf(t)$ when you are given the Laplace transform of its derivative (e.g., given $L{f}^{\prime \prime }(t)={\tan }^{-1}(1/s)$) and the initial conditions. [PYQ: 2016]
• Explicitly draw the circuit diagram from the word problem description before solving for the charge/current using Laplace Transforms. [PYQ: 2017]

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👨‍🏫 Instructor 2: Vector Calculus & Fourier Series

Chapter 8 & 9: Vector Differentiation & Geometry

• Determine unknown constants (e.g., $a,b,c$) for a scalar field so that its directional derivative has a specified maximum magnitude in a specific axis direction (e.g., parallel to the z-axis). [PYQ: 2015]
• Prove that specific electromagnetic field vectors ($E$ and $H$) satisfy the Wave Equation (${\nabla }^{2}u=\frac{{\partial }^{2}u}{\partial t^2}$) using their given curl and divergence properties. [PYQ: 2015]
• Calculate the equation of the Osculating Plane for a particle moving along a parametric path at a given time $t$. [PYQ: 2021]
• Sketch the rough path of a moving particle based on its parametric equations and physically comment on its movement. [PYQ: 2021]

Chapter 10: Vector Integration

• Calculate the total “Flux of Heat” out of a specific closed boundary (like a sphere) when the heat flow is defined by the gradient of a temperature scalar field ($\bar{F}=-\nabla T$). [PYQ: 2020]

Chapter 11: Fourier Series

• Write the general formula for Fourier coefficients and the Fourier series for a function with an arbitrary period $T=2L$ (e.g., $T=6$), rather than just standard $2\pi$ periods. [PYQ: 2016, 2019]
• Construct a piecewise mathematical function strictly from looking at a given graphical wave/plot before computing its Fourier series. [PYQ: 2022]
• Find the Fourier series of specific engineering waveforms, specifically the Half-wave rectified sinusoidal wave and the Alternating current after passing through a rectifier. [PYQ: 2017, 2019]
• Sketch the graph of a given periodic function or its resulting Fourier series over a specified interval. [PYQ: 2017]