Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate ↓


Algebra


Algebra is an essential field in mathematics that deals with the study of mathematical symbols and the rules for manipulating these symbols. In a broad sense, it is about finding unknowns or inserting real-life variables into equations and then solving them. When learning algebra, you will develop a good understanding of how to balance equations, work with variables, and solve an assortment of mathematical problems.

Understanding variables and constants

In algebra, we use letters like x, y and z to represent variables. A variable is a symbol used to represent a number that we don't know yet. Constants, on the other hand, are fixed values, like 3, 15, or -7.

Example: 2x + 5 = 15

In this instance:

  • 2 and 5 are constants.
  • x is a variable.

Operations in algebra

The basic operations in algebra include addition, subtraction, multiplication, and division. As in arithmetic, expressions and equations can be formed by applying these operations to variables and constants.

Addition and subtraction

Consider adding and subtracting terms:

Example: 3x + 4y - x + 2 = 10

The goal here is to simplify the expression by combining like terms. Like terms are terms in which the same variable is raised to the same power.

Simplified:

(3x - x) + 4y + 2 = 10
2x + 4y + 2 = 10

Multiplication and division

These operations apply to algebra in the same way as they do to numbers. Multiplying by a variable expands an equation, and dividing by a variable reduces it. It works like this:

Example: 5x * 2 = 10x

Division is often present when you solve for a variable. To isolate the variable, you may need to divide both sides of the equation:

Example: 6x = 18

To find x:

x = 18 / 6
x = 3

Solving algebraic equations

The purpose of solving equations is to find the unknown value, usually represented by a variable. There are different types of equations in algebra, including linear, quadratic, and polynomial equations. Let's start with the simplest form:

Linear equations

A linear equation is an equation in which the unknown value is not raised to a power other than one. It is often in the format ax + b = c. Here is a step-by-step example of solving a linear equation:

Equation: 4x - 7 = 5

1. First, isolate x by adding 7 to both sides:

4x - 7 + 7 = 5 + 7
4x = 12

2. Next, divide both sides by 4 to find the value of x:

x = 12 / 4
x = 3

Quadratic equations

Quadratic equations are equations in which a variable is raised to the second power, usually of the form ax^2 + bx + c = 0. Solving these equations usually involves factoring, the quadratic formula, or completing the square.

An example of factoring:

Equation: x^2 - 5x + 6 = 0

Factoring gives us:

(x - 2)(x - 3) = 0

Setting each bracket to zero gives the solution:

x - 2 = 0 => x = 2
x - 3 = 0 => x = 3

Using the quadratic formula

When factoring isn't easy, the quadratic formula is a reliable method:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

Consider the quadratic equation given earlier:

Equation: x^2 - 5x + 6 = 0

Here, a = 1, b = -5, and c = 6:

x = (5 ± sqrt((-5)^2 - 4*1*6)) / (2*1)
x = (5 ± sqrt(25 - 24)) / 2
x = (5 ± sqrt(1)) / 2

At the end:

x = (5 ± 1) / 2
x = 3 or x = 2

Polynomial equations

Polynomial equations proceed from quadratic equations, taking the form an*x^n + a(n-1)*x^(n-1) + ... + a1*x + a0 = 0. The complexity increases with the degree of the polynomial.

Example: 2x^3 - 3x^2 + x - 5 = 0

Solving high-degree polynomials often requires advanced techniques or methods of calculus, such as numerical solution.

Graphical representation

Graphs provide a visual representation of equations and inequalities. They give information about the nature of roots and the behavior of functions.

Graphs of linear equations

For the linear equation y = 2x + 3, plotting a graph means finding the value of y when the x value is given (and vice versa):

y = 2x + 3

The line shown shows that as x increases, y increases twice as fast, which shows a positive slope.

Graph of quadratic equations

Quadratic equations form parabolas. The graph of y = x^2 - 5x + 6 looks like this:

y = x^2 - 5x + 6

The parabola opens upward, and the points where it intersects the x-axis represent the roots of the quadratic equation.

Inequality

Inequalities are expressions that use <, >, <=, or >= instead of an equals sign. Solving them involves the same steps as equations, but the solutions are boundaries or intervals:

Example: 3x + 2 > 5

1. Subtract 2 from both sides:

3x > 3

2. Divide by 3:

x > 1

The solution is that x is greater than 1.

Graphing inequalities

The graphs of inequalities are different because they show a boundary or shaded area rather than a line or curve. For example, seeing that y < x + 2 involves shading below the line y = x + 2:

y = x + 2 y < x + 2

The shaded area shows all possible solutions to the inequality.

Functions and mapping

Algebra expands into functions, which are relations that uniquely associate members of one set with members of another set. A function f(x) tells how each input x is related to the output.

Consider f(x) = 2x. For each input, multiply by two to find the output.

Simple example:

  • If f(1), then f(1) = 2
  • If f(3), then f(3) = 6

Domain and range

The domain of a function is the entire set of possible input values, while the range is the set of possible output values.

For f(x) = x^2:

  • Domain: all real numbers
  • Range: all real numbers ≥ 0

Graphing functions

Graphing a function gives a visual representation of how f(x) relates to x.

f(x) = x²

The resulting curve, a parabola, visually represents the output of f(x) = x^2 for each input x.

Conclusion

Algebra is a vast and foundational part of mathematics that provides tools for problem-solving in many fields such as engineering, science, economics, and more. With algebra, you can create equations representing real-world problems, develop solutions, and build models describing complex systems.

The ability to understand and apply algebra is vital to advanced study in mathematics and its applications in technology and other sciences. Representing algebraic concepts through graphs enhances understanding by providing concrete connections to abstract equations and functions.


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