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Table of contents
1.
Introduction
2.
Definition
3.
Checking if linear transformation or notLet’s check if f(x,y)=(2x+1, y, x+y)= is a linear transformation or not.
4.
FAQs
5.
Key Takeaways
Last Updated: Mar 27, 2024
Easy

Linear Transformations

Author Malay Gain
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Ashwin Goyal
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Introduction

In Linear Algebra, linear transformation is widely used for matrix transformation. In vector algebra, we also apply the concept of the linear transformation in vector space. Transformation means nothing but a function where

is the domain of the function and is codomain. In this article, we will limit our discussion within matrix transformation.

Definition

Linear transformation is a function that satisfies the following properties :

  • T(X + Y) = T(X) + T(Y)
  • T(aX) = aT(X)

 

Where and

 

In case of matrix transformation, X and Y are nothing but matrices.

 

In another way, the function is linear transformation if we can associate some matrix with T(x) and each term of each component of T(x) is a number times one of the variables. 

 

For example,

The function f(x,y) = (2x, 3y) and g(x,y,z) = (z, 3+y, 0.5x) are linear transformation but  f(x,y) = (2x, y^2) not a linear transformation.

 

f(x,y) = (2x+1, y, x+y) is a linear transformation from

 

It can be also expressed as f(x,y)= (in matrix form)

Checking if linear transformation or not
Let’s check if f(x,y)=(2x+1, y, x+y)= is a linear transformation or not.

 

 

Proof

let……

X=, Y=

f(x1,y1) =f(x2,y2) =

 

f(X+Y) = = f(X) + f(Y) 

 

So, it satisfies the property of linear transformation. 

 

Hence, f(x,y) = (2x+1, y, x+y) is a linear transformation associated with matrices.

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FAQs

 

  1. What is the property of linear transformation in linear algebra?

Linear transformation is a function that satisfies the following properties :

  • T(X + Y) = T(X) + T(Y)
  • T(aX) = aT(X)

 

Where and

 

In case of matrix transformation, X and Y are nothing but matrices.

 

Key Takeaways

This article covered the properties of linear transformation and linear transformation associated with matrices.

 

Check out the Coding Ninjas Studio library for getting a better hold of the data structures and algorithms.

Check out this problem - Matrix Median

Side by side, you can also practice a wide variety of coding questions commonly asked in interviews in Coding Ninjas Studio. Along with coding questions, you can also find the interview experience of scholars working in renowned product-based companies here. 

 

Happy learning!

 

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