Basic concepts Flashcards
learn the basic concepts
Determinant
Scalar quantity obtained by the sum of products of the elements of a square matrix.
Example:
C = [ 8 6 ]
[ 3 4 ]
|C| = | 8 6 |
| 3 4 |
|C| = ( 8 x 4 ) + ( 6 x 3 ) = 32 + 18 = 14
Derivative
sensitivity of a functions output based on its input.
f’(x) = limh→0 [(f(x + h) - f(x))/h]
it tells you the rate at which something is changing at any given point.
Slope
change in y/change in x = delta y / delta x
Differentiation
The process of finding derivatives is called differentiation.
Limit
a limit is the value that a function approaches as the argument approaches some value.
Function
A function definition provides a rule or relationship that assigns each input exactly one output. In mathematics, it describes how inputs (from a domain) are mapped to outputs (in a range)
f(x) is dependent variable, x is the independent variable
Domain of a function
The domain of a function is the set of all possible input values (independent variable, typically
𝑥) for which the function is defined. In other words, it’s the set of all x-values that you can plug into the function without causing undefined behavior (like dividing by zero or taking the square root of a negative number in real numbers).
Range of a function
The range of a function is the set of all possible output values (dependent variable, typically f(x)) that the function can produce. It depends on the domain and the rule of the function.
For f(x)=x^2, the range is f(x) ≥ 0 because squaring any real number always gives a non-negative result.
Difference between domain and range
Domain restricts the inputs you can use.
The domain is all about the inputs (x) that the function cannot accept because they make the operation undefined. These restrictions typically arise due to:
- division by zero
- square roots of negative numbers
- logarithms of non-positive numbers
The Range is all about the outputs f(x)) that the function cannot produce because the calculations or rules of the function make them impossible. These restrictions are a consequence of:
- The nature of the operation (e.g., a square root can never produce a negative output)
- Limits of the function’s behavior (e.g., f(x)= 1/x can never equal zero because the numerator is constant).
Logarithm
A logarithm is the inverse operation of exponentiation. It answers the question: “To what power must a base be raised to produce a given number?”
Why are logarithms useful for calculus
Logarithmic functions are important in calculus, particularly for:
Integration: The integral of 1/x is ln|x|, and logarithms are used to solve many integrals involving powers of x.
Differentiation: The derivative of ln(x) is 1/x, which is fundamental in optimization and other areas.
Example: Differentiating a logarithmic function like y = ln(x²) can help you find the rate of change.
Integration vs differentiation
Integration is the reverse of differentiation. It is the process of finding the accumulated area under a curve or the total quantity over an interval. While differentiation breaks things into rates of change, integration adds up those changes to find the total.
Intuition: If differentiation is like finding the speed of a car at any moment, integration is like finding the total distance the car has traveled over a period of time.
The integral of a function f(x) is often written as ∫ f(x) dx, where dx represents a small change in x.
If f(x) is a function, the derivative of f(x) is often written as f’(x) or (d/dx) f(x).
Eigendecomposition
Eigendecomposition is a method or process used to calculate the eigenvalues and eigenvectors of a square matrix.
Here’s a breakdown of what eigendecomposition does:
It finds the eigenvalues (λ) and eigenvectors (v) of a matrix.
Once these eigenvalues and eigenvectors are found, the matrix can be written in a decomposed form:
A = V Λ V⁻¹
where:
A is the original matrix,
V is the matrix of eigenvectors,
Λ is the diagonal matrix with the eigenvalues on its diagonal, and
V⁻¹ is the inverse of the matrix V.
Inverse of a matrix
The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix.
In other words, for a square matrix A, the inverse of A, denoted as A⁻¹, satisfies the following equation:
A ⋅ A⁻¹ = A⁻¹ ⋅ A = I
Where I is the identity matrix (a matrix with ones on the diagonal and zeros elsewhere).
Key Points:
Existence: Not all matrices have an inverse. A matrix has an inverse only if it is square (same number of rows and columns) and non-singular (its determinant is not zero). If the determinant of a matrix is zero, it is called a singular matrix, and it does not have an inverse.
Identity Matrix: The identity matrix I is a special square matrix where all the diagonal elements are 1, and all other elements are 0. It behaves like “1” in matrix multiplication, similar to how 1 works in regular arithmetic.
For example, the 2x2 identity matrix is:
I = [ 1 0 ]
[ 0 1 ]
Multiplication: The product of a matrix A and its inverse A⁻¹ is the identity matrix. This means that multiplying a matrix by its inverse “undoes” the operation of the matrix.
Identity matrix
Matrix with ones on the diagonal and zeros elsewhere
The identity matrix is a special type of square matrix that plays a similar role to the number “1” in regular arithmetic. It is used in matrix multiplication and has unique properties that make it very useful in linear algebra.
Square Matrix: The identity matrix is ALWAYS SQUARE, meaning it has the same number of rows and columns.
