Basic, Advanced, and Linear Algebra Review

Key mathematical vocabulary, formulas, properties, and definitions covering basic, advanced, and linear algebra concepts.

Domain of a Function

The set of all possible input values (independent variable) for which the function is defined, representing all the $x$-values for which the function produces meaningful output.

Range of a Function

The set of all possible output values (dependent variable) that a function can produce, representing all the $y$-values that the function can attain based on its domain.

Two Golden Rules for Finding the Domain

1. No Division by Zero: Ensure the denominator of a fraction is never zero. 2. No Square Roots of Negative Numbers: Values inside a square root must be non-negative.

Arithmetic Sequence

A sequence of numbers in which the difference between any two consecutive terms is constant.

Common Difference ($d$)

The constant difference between any two consecutive terms in an arithmetic sequence.

$n\text{th}$ Term of an Arithmetic Sequence

The formula given by $a_n = a_1 + (n - 1)d$, where $a_n$ is the $n\text{th}$ term, $a_1$ is the first term, $d$ is the common difference, and $n$ is the position of the term.

Geometric Sequence

A sequence of numbers (also known as a geometric progression) where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Common Ratio ($r$)

The fixed, non-zero number multiplied by the preceding term to generate each successive term in a geometric sequence.

$n\text{th}$ Term of a Geometric Sequence

The formula given by $a_n = a_1 \cdot r^{(n - 1)}$, where $a_n$ is the $n\text{th}$ term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the position of the term.

Determinant of a $2 \times 2$ Matrix

A special number calculated from the elements of a square matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ using the formula $\text{Determinant} = ad - bc$.

Trace of a Matrix ($\text{tr}(A)$)

The sum of the diagonal elements of a square matrix from the top-left to the bottom-right.

Logarithmic Function

The inverse of an exponential function, representing the exponent to which a given base ($b$) must be raised to obtain a specific number, with general form $y = \log_b(x)$.

Characteristic (of a Logarithm)

The integer or integral part of the logarithm (base $10$) of a number; if the number is written in scientific notation, it corresponds to the exponent of the power of $10$.

Mantissa (of a Logarithm)

The decimal or fractional part of the logarithm following the decimal point.

Product Rule of Logarithms

The property stating that the logarithm of a product is equal to the sum of the logarithms of each factor: $\log_a(xy) = \log_a(x) + \log_a(y)$.

Quotient Rule of Logarithms

The property stating that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor: $\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$.

Power Rule of Logarithms

The property stating that the logarithm of a power equals the exponent multiplied by the logarithm of the base: $\log_a\left(x^p\right) = p \cdot \log_a(x)$.

Change of Base Rule

The logarithmic property used to rewrite a logarithm in terms of a different base: $\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$.

Equality Rule of Logarithms

The property stating that if $\log_a(x) = \log_a(y)$, then $x = y$.

Roots (Zeroes or Solutions) of an Equation

The values of the variable that make an algebraic or quadratic equation equal to zero.

Function Composition ($(f \circ g)(x)$)

An operation where the output of function $g(x)$ is used as the input for function $f(x)$, expressed as $f(g(x))$.