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 |
Also, Check
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.
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