Mathematics

Matrix Transformations – Linear Algebra

by ComputingNotes
1. Functions and Transformations In standard algebra, a function takes a single number as an input and outputs a new number: f(x) = y. In Linear Algebra, we scale this concept up. A Transformation (often denoted as T) is a mathematical rule that takes an entire Vector as an input and assigns it to a ... Read More

Algebraic Properties of Matrices

by ComputingNotes
1. Properties of Matrix Arithmetic Matrix arithmetic shares many rules with standard algebra, but there are crucial differences. Here are the core laws: Commutative Law for Addition: A + B = B + A Associative Law: A + (B + C) = (A + B) + C A(BC) = (AB)C Distributive Law: A(B + C) ... Read More

Matrices and Matrix Operations – Linear Algebra

by ComputingNotes
1 & 2. Matrix Notation, Terminology & Equality A Matrix is simply a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices are considered Equal only if they have the exact same dimensions AND every corresponding element is identical. 2 4 -1 9 = 2 4 -1 9 Conversely, here is ... Read More

Calculus – $a^x$ and $log_a x$

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In calculus, exponential functions with a base other than e (like ax) and logarithms with base a (like loga x) are handled by converting them into their “natural” forms. Since the natural exponential ex and natural log ln(x) have very simple derivatives, we use them as the bridge to understand ax and loga x. Interactive ... Read More

Caculus – Exponential Functions

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Resource: The Exponential Function Suppose we have a quantity \(y\), whose rate of change over time is proportional to the amount present. We can describe this relationship using a differential equation: \[ \frac{dy}{dt} = ky \] If we define the initial state where \(y = y_0\) at time \(t = 0\), the solution to this ... Read More

Systems of Linear Equations and Matrices

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References: Linear Equation: At its simplest, a linear equation is an algebraic equation that creates a straight line when plotted on a graph. Every variable in the equation is raised to the first power (meaning no exponents like $x^2$ or $y^3$), and there are no variables multiplied by each other. The Standard Forms Depending on ... Read More

Calculus – Natural Logarithms

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Resources: What is Natural Logarithm? In calculus, the natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.71828. Unlike base-10 logarithms (common logs), the natural logarithm, denoted as $ln(x)$, is preferred in calculus because it simplifies the differentiation and integration of exponential functions. 1. ... Read More

Euler’s Number $e$

by ComputingNotes
Resources: What is Euler’s Number? Euler’s number (eā‰ˆ2.71828e is approximately equal to 2.71828) was first discovered by Swiss mathematician in Jacob Bernoulli in1683 while studying compound interest. However, it is named after Leonhard Euler, who popularized the constant, discovered its connection to calculus, calculated its value to 23 decimal places, and used the letter š‘’ ... Read More

Calculus – Inverse Functions

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Resources: What is Inverse Function in Calculus? In calculus, an inverse function is a function that “undoes” the action of another function. If a function $f$ takes an input $x$ and gives an output y, its inverse $f^{āˆ’1}$ takes $y$ and brings you back to $x$. Mathematically: $f(x) = y \iff f^{-1}(y) = x$ 1. ... Read More

Calculus – Transcendental Functions

by ComputingNotes
References: Transcendental Functions In mathematics, a transcendental function is a function that does not satisfy a polynomial equation with polynomial coefficients. To put it simply: it is any function that is not algebraic. While algebraic functions can be constructed using a finite number of elementary operations (addition, subtraction, multiplication, division, and taking roots), transcendental functions ... Read More
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