Vector and Matrix are Mathematical quantities used in linear algebra. Vector is a quantity that includes magnitude and direction like velocity, displacement. They are used in three-dimensional problems to make them simplified. Matrix is a rectangular array of numbers used in linear algebra for making linear transformations and linear equations.
Vector vs Matrix
The main difference between Vector and Matrix is that Vector is an array of numbers with a single index, whereas Matrix is a rectangular array of numbers with two indices as row and column. Vector is a way to simplify three-dimensional figures in geometry, whereas Matrix is used in linear algebra for linear transformations.
Vector is a quantity that is represented by a letter with an arrow on top to distinguish it from other mathematical quantities. It represents magnitude and direction. It is an array of numbers called elements in a Vector. They are enclosed in open brackets.
Matrix is denoted by upper case letters in brackets or parentheses. It is a rectangular array of numbers with elements or entries in it. It has a row vector and column vector which forms a matrix. It has two indices indicating row and column. Matrix extends to its higher dimension in linear algebra.
Comparison Table Between Vector and Matrix
| Parameters of Comparison | Vector | Matrix |
| Definition | Vector is an array of numbers with elements enclosed in open brackets. | Matrix is a rectangular array of elements or entries in a row and column vector enclosed in open brackets. |
| Represents | A Vector represents magnitude and direction in its quantity with units. | A Matrix represents the linear transformations and coefficients of the linear equations. |
| Index | Vector has its elements in a single index. | A Matrix has its elements or entries in two indices denoted as row x column. |
| Denoted | A Vector is denoted in letters with an arrow on top of it to differentiate it from other quantities. | A Matrix is denoted in upper case letters. |
| Uses | A Vector is used in simplifying three-dimensional objects in geometry. | A Matrix is used in linear algebra for linear transformations and forming linear equations. |
What is Vector?
Vector is defined as a quantity of an object that has both magnitude and direction. It is denoted by a letter with an arrow on it. If there exist two vectors they are the same if their magnitude and direction are equal. Magnitude specifies the size of the vector and direction specifies the direction in which the object is in motion.
Vector is very important in mathematics, and physics in various domains like linear algebra. A vector can be combined with another vector with its head attached to the other vector’s tail. Vector is represented by the letter of the endpoints of direction with the direction arrow placed above the letter.
Vector is not restricted to mathematical operations. The addition of two or more vectors results in the same magnitude and direction according to the cumulative and associative law which is the same for the subtraction of vectors as well. In scalar multiplication with a vector, the magnitude gets changed whereas the direction remains the same.
Vector can be used to find the direction of the motion of the object, how gravity is implied on an object, used in oscillators, quantum mechanics, fluid mechanics, in the theory of relativity, the motion of an object across a plane, is used in wave propagation, sound propagation helps in determining the force applied in a three-dimensional object.
What is Matrix?
A matrix is a rectangular array of numbers or elements or entries arranged in rows and columns. They are denoted by letters written in upper case. The order of a Matrix is defined in the number of rows and columns. The numbers in the matrix are known as entries and each entry is known as an element.
A Matrix in its plural form is known as Matrices. The size of the matrix is indicated as row x columns, which is written as n x m where n denotes rows and m denotes columns in the matrix. Various operations are performed with two or more matrices and that helps in finding the determinant of the matrix which ends up to be the scalar quantity of the equation.
A matrix that has all its elements as zero is known as a Zero matrix or Null matrix. If the elements above or below the principal diagonal of a square matrix are zero is known as a triangular matrix, if the elements below the principal diagonal are zero then it is known as Upper Triangular Matrix, if the elements above the principal diagonal are zero then it is known as Lower Triangular Matrix.
The matrix in which the principal diagonal elements are one is known as Identity Matrix. The matrix in which the number of rows is greater than the number of columns is known as Vertical Matrix, if the number of columns is greater than the number of rows then it is called a Horizontal Matrix.
Main Differences Between Vector and Matrix
- Vector is a quantity that has magnitude and direction of the object in motion, whereas Matrix is the rectangular array of elements which are either zero or non-zero and parameters to be estimated if necessary.
- Vector is denoted as letter or letters with arrow indicating direction of the motion of the object, whereas a Matrix is one which is denoted as upper case alphabets.
- Vectors are array of numbers which are used in geometry for simplifying three-dimensional objects to determine the direction of the motion of the objects, whereas a Matrix is one which plays an important role in Linear Algebra for linear transformations and coefficients of the linear equations in linear models.
- Vectors have single index in the rectangular array, whereas Matrix has two indices in their formation.
- Vectors do not change in their magnitude and direction in their mathematical operations, whereas a Matrix change in their magnitude in mathematical operations such as multiplicative operation with respect to associative and cumulative laws.
Conclusion
Vector and Matrix are used in mathematics, and physics for transformations and equations. Vectors are used in physics to determine the direction and gravity of the objects in Quantum Mechanics and Fluid Mechanics. Matrix is used in Linear models and transformation in principal component analysis.
Vector is an array of numbers or elements enclosed in an open bracket that represents the magnitude and direction of the object. Matrix is a rectangular array of numbers in the form of rows and columns with entries and elements representing the magnitude and in mathematical operations they form determinants.
References
- https://www.sciencedirect.com/science/article/pii/004269899400257M
- https://pubs.acs.org/doi/pdf/10.1021/ie50548a027