Is Linear Algebra the Same as Matrix Algebra?
Is Linear Algebra the Same as Matrix Algebra?
No, linear algebra is not the same as matrix algebra. Linear algebra talks about linear spaces, and related concepts.
Whereas matrices and vectors represent linear spaces and actual spaces. They represent appropriate entities only when the bases are chosen.
Linear algebraic techniques become useful even when particular representation is not given.
Linear algebra
Linear mathematics is that branch of mathematics which talks about linear equations and how they are represented in the vector space using matrices. In other words, linear algebra deals with the study of linear functions and vectors. It is one of the most important topics of mathematics. Most modern geometrical concepts are derived from linear algebra.
Linear algebra helps in the modeling of many natural phenomena. That is why, it is an important part of engineering and physics. Linear equations, vector spaces and matrices are the most important part of this subject.
What is linear algebra?
Linear algebra can besaid to be that branch of mathematics which covers the study of linear functions in vector spaces. When information related to linear functions is given in a well-planned manner, then it results in a matrix. Therefore, linear algebra deals with vector spaces, vectors, linear functions, matrices and the system of linear equations. These mathematical topics are essential for topics such as geometry and functional analysis.
Branches of linear algebra
Linear algebra can be divided into three branches based on the level of difficulty and the kind of topics that are within it. They are elementary, advanced, and applied linear algebra. Each branch deals with different aspects of matrices, vectors, and linear functions.
Elementary linear algebra
Elementary linear algebra talks about the basics of linear algebra. It covers simple matrix operations, various computations that can be calculated on a system of linear equations, and some aspects of vectors. Some important terms of elementary linear algebra are as follows:
Scalars- A scalar is quantity which have only magnitude and no direction. It is an element which defines a vector space. In linear algebra, scalars are real numbers.
Vectors- A vector can be defined as an element in a vector space. It is a quantity which talks about both the direction and magnitude of an element.
Vector space- The vector space has vectors that can be added together and multiplied by scalars.
Matrix- A matrix is a rectangular arrangement of rows and columns.
Marix operations- They are simple arithmetic operations such as addition, subtraction, and multiplication that can be done on matrices.
Matrix algebra
The plural form of matrix is matrices. It is a rectangular arrangement or a table where numbers or elements are arranged in rows and columns. There is no limit to the number of columns and rows in matrices. Various operation can be done on matrices- matrix addition, scalar multiplication, matrix multiplication, transposition, etc.
There are certain rules to be followed while doing matrix operation, such as matrices can be added or subtracted when there are same number of rows and columns and they can be multiplied when the columns in the first and rows in the second are the same.
What are matrices?
Matrices are the rectangular array of numbers, variables, symbols, or expressions in the rectangular table which have many numbers of rows and columns. The numbers or entries in the matrix called as elements. Horizontal entries for matrices are known as rows and the vertical entries are called as columns.
Notation of matrices
When a matrix has m rows and n columns, then it will contain m * n elements. A matrix is shown by the uppercase letter, in this case ‘A’, and the elements in the matrix are shown by the lower-case letter and two subscripts showing the position of the element in the number of row and column in the same order, in this case 
, where i stands for the number of rows, and j is the number of columns. Let us take an example. In the given matrix A, element in the 3rd row and 2nd column will be 
, can be verified in the matrix as shown below:
How to solve matrices?
We can solve matrices by doing operations on them like addition, subtraction, multiplication, and so on.
Do you know the answer of the below question:
Q: If A=[(1 ,0, 0),(0, 1,0),( a,b, -1)] , then A^2 is equal to
(a) a null matrix
(b) a unit matrix
(c) -A
(d) A
How to solve system of equations using matrices?
Suppose there are two matrices A and B where A is the coefficient matrix and B is the constant matrix. There is a third matrix X which have all the variables of the equations. This matrix is called as a variable matrix. Matrix A is of the order m * n, and B is the column matrix of the order m* 1. The product of matrix A and matrix X is matrix B. Thus, X is a column matrix as well of the order n * 1.