Table of Linear Algebra Symbols

dot

scalar product

a.b

cross

vector product

a x b

tensor product

tensor product of A and B

A B

Square Brackets

Square brackets, meant by the symbols “[ ]”, fill different needs in various contexts, including arithmetic, programming, semantics, and writing conventions

x has a place with the closed interval from 3 to 7, including the both endpoints. This is indicated x as ∈ [3,7].

Curly Brackets or Set Symbol

Used to group components.

5 × { 4 + 5 }

Here, the curly brackets indicates that the addition operation inside should be perform before multiply with 5

If set A is a set of first 3 natural numbers then A = {1, 2, 3 }

Parentheses

Indicate the request for tasks

(4 + 4) × 3

Here, Parentheses indicates that addition operation should be perform before multiplying.

Set A

Set is always denoted by a capital letter in curly bracket

If set A is set of even numbers then

{A} = {2, 4, 6, 8…}

Subset

Subset means all element of a set is member of another set

A = {1, 3, a}

B = {a, b, 1, 2, 3, 4, 5}

A ⊆ B

Proper Subset

It represents all of the elements of set A are present in set B and set A is not equal to set B.

A = {1, 2, a}

B = {a, b, c, 2, 4, 5, 1}

A ⊂ B

Not a Subset

It determines A is not a subset of set B.

A = {1, 2, 3}

B = {a, b, c}

A ⊄ B

Superset

Superset means the set on left side of symbol has all the members of another set

Integeer is superset of Natural Number

Empty Set

It determines that there is no element in a set.

{ } = Ø

Power Set

It is the set that contains all possible subsets of a set A, including the set itself and the null set.

If A = {a, b}

P(A) = {{ }, {a}, {b}, {a, b}}

Union of Set

It means combining elements of two sets whiling keeping the common elements only once

A = {2, 3, 4}, B = {2, 4, 6}

Then A ⋃ B = {2, 3, 4, 6}

Intersection of Set

Intersection of sets means fidning out common elements between two sets

A = {2, 3, 4}, B = {2, 4, 6}

Then A ⋂ B = {2, 4}

Cardinality of Set

It denotes the number of elements in a given set

A = {2, 4, 6} then n(A) = 3

Null Set

Null set means there is no element in that set

Set of Natural Number greater than 2 but less than 3

Set of natural numbers

Positive integers (including zero)

0, 1, 2, 3, …

Set of integers

Whole numbers (positive, negative, and zero)

-3, -2, -1, 0, 1, 2, 3, …

Set of real numbers

All rational and irrational numbers

π, e, √2, 3/2, …

Matrices

Matrix is represented by Capital letters, matrices are arrays of numbers, symbols or expressions

A_3×2 =[Tex] \begin{bmatrix} 1& 2\\3 & 4 \\ 5& 6\\ \end{bmatrix}[/Tex]

Determinant

It represents the Determinant of a square matrix A.

If we have matrix A =[Tex] \begin{bmatrix} 1& 2\\ 3& 4\\ \end{bmatrix}[/Tex]then |A| = |1 × 4 – 2 ×3| = 4 – 6 = -2

Tanspose of Matrix

In Transpose of Matrix, the elements of rows are arranged in column and vice versa

If we have matrix A =[Tex] \begin{bmatrix} 1& 2\\ 3& 4\\ \end{bmatrix}[/Tex] then A^T =[Tex]\begin{bmatrix} 1& 3\\ 2& 4\\ \end{bmatrix}[/Tex]

Inverse of Matrix

Inverse of Matrix basically means finding a matrix that when multiplied to orginal matrix give

For Matrix A =[Tex] \begin{bmatrix} 1& 2\\ 3& 4\\ \end{bmatrix}[/Tex]A^-1 = [Tex] \begin{bmatrix} -2& 1\\ 3/2& -1/2\\ \end{bmatrix}[/Tex]

Hermitian Matrix

In Hermitian Matrix we find matrix conjugate transpose

For matrix A, A = AH where AH is the conjugate transpose of matrix A.

(A^*)_ij =(A^T)_ji

matrix rank

rank of matrix A

rank(A) =3

dimension

dimension of matrix A

rank(U) =3

addition

It combines two or more values.

Solve 5 + 5

solution = 10

subtraction

It finds the difference between the two values.

Solve 7- 5

solution = 2

multiplication

It multiplies two or more values.

Solve 5 x 5

solution = 25

division

Represents sharing or dividing.

Solve 10 / 5

solution = 2

Equal to

Indicates correspondence between two expressions.

5 + 5 = 10 Here ,The equal sign denotes that the sum of 5 and 5 is equal to 10

Not equal to

Demonstrates inequality.

5 ≠ 3

The not equal to sign indicates that 5 is not equal to 3

Less than

This less than symbol (<) is a principal mathematical symbol used to denote that one amount is smaller than another.

12 < 15

Solution : True, Because 12 is less than of 15

Greater than

This Greater than symbol (>) is a principal mathematical symbol used to denote that one amount is Greater than another

15 > 5

Solution : True, Because 15 s greater than of 5

Less than or equal To

This ( ≤ ) symbol represent to less than or equal to. This is used to express that one value is less than or equal to another.

x ≤ 5

Here x is less than or equal to 5

Greater than or Equal To

This ( ≥ ) symbol represents to Greater than or equal To. This is used to express that one value is greater than or equal to another.

x ≥ 6

Here x is greater than or equal to 6

Much less than

Value on left side of the symbol is much less than value on right side

1 ≪ 100

It means 1 is much less than 100

Much greater than

Value on left side of the symbol is much greater than value on right side

1 ≫ 100

it means 1 is much greater than 100

Directly Proportional

It means increase in value of one quantity will lead to increase in value of other quantity

Total Bill increases if you buy more product. Hence, total bill is directly proportional to number of objects

modulus

It represent absolute value of x

It finds the modulus of x