Linear Algebra Symbols

Linear Algebra Symbols are mainly focused on comprehending how various systems of linear equations behave and may be solved. This is performed by storing the equations in matrices and vectors, both of which are mathematical objects that may be handled in several ways.

In this article, we will learn about Linear Algebra Symbols their definition and meaning, and others in detail.

What is Linear Algebra?

Linear algebra is a branch of algebra, find uses in both pure and practical mathematics. It deals with the linear mappings of the vector spaces. It also involves learning about lines and planes. It is the study of linear systems of equations with transformational features. It is used in almost all areas of mathematics. It deals with the representation of linear equations for linear functions in matrices and vector spaces.

Linear Algebra Symbols in Maths

Linear Algebra symbols in maths are the unique characters that have their specific meaning in a mathematical operation. Any Linear algebraic expression mainly consists of variables and constants. Let’s learn first what is the symbol of variable and constant.

Variable Symbol

Variable as the name suggests has no fixed value and their value change in different situations. Variables in Algebra are represented by Alphabets such as A, B, C… or a, b, c…. and by Greek Letters such as α, β, γ, etc. The unknown angle is represented by θ

Constant Symbol

Constant are those which have fixed value. Constant in algebra are represented by Numbers in maths such as 1, 2, 3, -1, -2…. Greek letters such as pi(π) is also a constant whose value is approximately equal to 3.14, and euler’s number ‘e’ whose value is equal to 2.71

Table of Linear Algebra Symbols

Linear Algebra includes the study of matrices, set theory, determinant etc. The below table shows the Linear Algebra Symbols and its name ,meaning and example for each.

SYMBOL	NAME	MEANING/DEFINITION	EXAMPLE
.	dot	scalar product	a.b
x	cross	vector product	a x b
A 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.
{A}	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
Ø	Empty Set	It determines that there is no element in a set.	{ } = Ø
P(X)	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}
n(A)	Cardinality of Set	It denotes the number of elements in a given set	A = {2, 4, 6} then n(A) = 3
Φ	Null Set	Null set means there is no element in that set	Set of Natural Number greater than 2 but less than 3
ℕ	Set of natural numbers	Positive integers (including zero)	0, 1, 2, 3, …
ℤ	Set of integers	Whole numbers (positive, negative, and zero)	-3, -2, -1, 0, 1, 2, 3, …
ℝ	Set of real numbers	All rational and irrational numbers	π, e, √2, 3/2, …
Aij	Matrices	Matrix is represented by Capital letters, matrices are arrays of numbers, symbols or expressions	A3×2 =[Tex] \begin{bmatrix} 1& 2\\3 & 4 \\ 5& 6\\ \end{bmatrix}[/Tex]
| A | or det(A)	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
AT	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 AT =[Tex]\begin{bmatrix} 1& 3\\ 2& 4\\ \end{bmatrix}[/Tex]
A-1	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]
A*	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 =(AT)ji
rank(A)	matrix rank	rank of matrix A	rank(A) =3
dim(U)	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
* or x	multiplication	It multiplies two or more values.	Solve 5 x 5 solution = 25
/ or ÷	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
|x|	modulus	It represent absolute value of x	It finds the modulus of x

Linear Algebra Symbols Solved Examples

Example 1: Find the sum of the two vectors [Tex]\overrightarrow{\rm A}[/Tex]= 3i + 4j + 5k and [Tex]\overrightarrow{\rm B}[/Tex]= -i + 2j + ?

Solution:

[Tex]\overrightarrow{\rm A} + \overrightarrow{\rm B}[/Tex]= (3-1)i + (2 + 4)j + (5 + 1)k = 2i + 6j + 6k

Example 2: Find the sum of the two matrices

[Tex] \begin{bmatrix} 3 & 3\\ 4 & 6\\ \end{bmatrix} and \begin{bmatrix} 1 &-2 \\ 3 & 4 \end{bmatrix}[/Tex] ?

Solution:

Let the two matrices be P = [Tex]\begin{bmatrix} 3 & 3\\ 4 & 6\\ \end{bmatrix}[/Tex]

Q = [Tex]\begin{bmatrix} 1 &-2 \\ 3 & 4 \end{bmatrix}[/Tex]

S = P + Q = [Tex]\begin{bmatrix} 3 & 3\\ 4 & 6\\ \end{bmatrix} + \begin{bmatrix} 1 &-2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 4 & 1\\ 7& 10\\ \end{bmatrix}[/Tex]

Example 3: For the equation 2x – 3 = 2, Solve for x ?

Solution:

6x – 3 = 2

⇒ 6x = 2 + 3

⇒ 6x = 5

⇒ x = 5 / 6 = 0.83

Example 4: Find the inersection of both A = {2, 3, 4}, B = {2, 4, 6,9,11}

Solution:

A = {2, 3, 4}, B = {2, 4, 6,9,11}

Then A ⋂ B = {2, 4}

Example 5: Given two values A = 5 and B = 20, Find the product of A and B

Solution:

Product of A and B = A × B = 5 × 20 = 100

Example 6: If the set A={3,5,7,9} then find n(A) ?

Solution:

n(A) denotes the number of elements in a given set, A={3,5,7,9} then n(A) = 4

Linear Algebra Symbols Practice Questions

Q1. Given two values A = 5 and B = 20, Find the division of A and B ?

Q2. Find the union of both A = {2, 3, 4,5}, B = {2, 4, 6,9,11} ,A ⋃ B ?

Q3. If the set A={3,5,7,9} then find P(A) ?

Q4. If the set A={3,5,7,9,13} then find n(A) ?

Q5. For the equation 6y – 5 = 20, Solve for y ?

Linear Algebra Symbols – FAQs

What does ∈ mean in Linear Algebra?

∈ is the symbol of Belongs To. The notation x∈A denotes that x is an element of the set A.

Why is Symbol usage necessary in Algebra?

Symbols makes Complex ideas simpler, easier to work with, and easier to understand by using symbols and notation.

What do the Greek Symbols mean in Linear Algebra

Scalars are represented by the Greek letters α, β, γ, λ, v, μ

What does ≠ Symbol mean in Linear Algebra ?

≠ symbol mean Not equal to in linear algebra, it Demonstrates inequality.

What does P(X) Symbol mean in Linear Algebra ?

P(X) symbol mean power set in linear algebra. It is the set that contains all possible subsets of a set A, including the set itself and the null set.

How to Represent Null Set?

In linear algebra an null set is represented by the symbol Φ, which means there is no element in that set.