Linear Algebra: Overview, Questions, Preparation

Linear algebra is the area of Mathematics where we study rotations in space, differential equations, equation of a circle, and other equation related questions. In this topic, you will learn that linear algebra is not a general algebra but vector spaces. 

How Does Linear Algebra Work?

The topics Vector Spaces, Linear Functions and, matrics are interrelated concepts. Vectors are the components that we can add, and linear functions are the components that consist of the addition of the vectors. 

The general linear equation represented below:
a1x1 + a2x2……….+anxn = b
Here,
a’s – represents the coefficients
x’s – represents the unknowns
b – represents the constant

Vectors Spaces

It has both physical quantities, direction, and magnitude. There are operations related to vector addition and scalar multiplication. These operations are vector axioms. So, vector axioms are: 
Commutative of addition: a + b = b + a
Associativity of addition: a + (b + c) = (a + b) + c
Additive identity: a + 0 = 0 + a = a
 Where 0 is an element in V called zero vector.
Additive inverse: a + (-a) + (-a) + a = 0
 a, -a belongs to V

Linear Function

It is a type of algebraic equation where every term is either a constant or product of constants or an independent variable power 1. In this topic, vectors form linear functions. 
According to the formula formation, the linear function is: 
A function L : Rn → Rm is linear 
When,
(i) L(x + y) = L(x) + L(y)
(ii) L(αx) = αL(x)
for all x, y ∈ Rn, α ∈ R

Linear Algebra Matrices 

Matrix is a type of linear function. It is the result of the arrangement of particular types of linear functions. Matrix is present in all types of Mathematics and its application and every formula that works in mathematical form. 

A is an m × n matrix, and then we get a linear function L: Rn → Rm by defining
L(x) = Ax
or 
Ax = B  
The definition of addition and multiplication is as follows: 

• Addition

Let A and B be matrices of the same size m×n over K. Then the sum A+B is defined by adding corresponding entries: 
                                         (A+B)i j = Ai j +Bi j

• Multiplication 

Let A be an m×n matrix and B an n× p matrix over K. Then the product AB is the m × p matrix whose (i, j) entry is obtained by multiplying each element in the ith row of A by the corresponding element in the jth column of B and summing: 
                            (AB)i j = n ∑ k=1 AikBk j. 

Note that we can only add or multiply matrices if their sizes satisfy appropriate conditions.  In particular, for a fixed value of n, we can add and multiply n × n matrices. 

The weightage of Linear Algebra in Class 10 boards is 8 marks. In Class 12, the weightage is 10 marks.  

1: Classify the following measures as scalars and vectors. (i) 5seconds (ii) 1000cm(iii) 10Newton (iv) 30km/hr (v) 10g/cm³ (vi) 20m/s towards north
Solution:

(i) Time-scalar (ii) Volume-scalar (iii) Force-vector (iv) Speed-scalar (v) Density-scalar (vi) Velocity-vector

2: Write a column vector in Kn. 
Solution:

It can be thought of as an n×1 matrix, while a row vector is a 1×n matrix. 

3:  What is the order of matrices with 8 elements? 
Solution:

1×8, 8×1, 4×2, 2×4

Q: Can we use the matrix as inputs?   

Q: Can we use the normal addition process? 

Q: How to write matrices in terms of associative multiplication? 

Q: How to write matrices in terms of distributive multiplication? 

Q: What is linear algebra?