1. LINEAR TRANSFORMATION
Presented By : Pin :
1. NAIK MUSAIB
ARMIET/CS24/NM251
2. PATIL ANIKET ARMIET/CS24/PA253
3. MASIR MOMIN ARMIET/CS24/
MM244
4. SHAIKH IBRAHIM
ARMIET/CS24/SI233
(Branch): COMPUTER
Semester-IV
2. Introduction
• A linear transformation is a special type of function in linear
algebra that transforms vectors from one space to another, in a
way that preserves the structure of the space.
• Imagine taking a shape made of arrows (vectors) and moving or
stretching it — but without bending or breaking it. That’s what
a linear transformation does.
3. What is Linear Transformation?
A linear transformation is a function between two vector spaces
that preserves:
1.Vector addition
2.Scalar multiplication
In simpler terms, a linear transformation doesn’t "distort" the
structure of the space—it keeps lines straight and scales/adds
vectors in a predictable way.
4. Why Do We Study Linear Transformations?
Linear transformations help us understand how shapes, systems, or data
change when we:
• Rotate, reflect, stretch, or compress them
• Switch coordinate systems
• Solve systems of linear equations
• Work in areas like computer graphics, machine learning, physics, and
more
5. Steps in a Linear Transformation
Step 1: Define the Transformation
Step 2: Represent It as a Matrix
Step 3: Write the Input Vector
Step 4: Multiply Matrix and Vector
Step 5: Interpret the Result
6. Mathematical Formulation
A linear transformation is a function, : ^ → ^
𝑇 𝑅 𝑛 𝑅 𝑚
that satisfies two key properties for all vectors , ^ ,and all scalars
𝑢 𝑣∈𝑅𝑛
𝑢𝑣
𝑢𝑣
𝑢𝑣
𝑢𝑣
𝑢𝑣
𝑢𝑣
𝑢𝑣
𝑢𝑣
𝑢𝑣
𝑢𝑣
𝑢𝑣
𝑢𝑣
𝑢𝑣 𝑢 𝑣
:
𝑐∈𝑅
1.Additivity (Preserves addition):
𝑇( + )= ( )+ ( )
𝑢⃗ 𝑣⃗ 𝑇 𝑢⃗ 𝑇 𝑣⃗
2. Homogeneity (Preserves scalar multiplication):
𝑇( ) = ( )
𝑐⋅𝑣 𝑐⋅𝑇𝑣
𝑣⃗ 𝑣⃗
If both properties are satisfied, is a linear transformation.
𝑇
7. Matrix Representation
• Any linear transformation : →
𝑇 𝑅𝑛 𝑅𝑚 can be represented by
an ×
𝑚 𝑛 matrix ,
𝐴 such that:
• 𝑇( )= T
𝑥⃗ 𝐴𝑥⃗
Where:
• 𝑥⃗is a column vector in ^ .
𝑅 𝑛
• 𝐴 is the matrix representing the transformation.
• 𝑇( )
𝑥⃗ is the transformed vector in ^ .
𝑅 𝑚
15. Conclusion
Linear transformations are fundamental tools in mathematics and its applications. They
provide a structured way to map vectors from one space to another while preserving the
core properties of linearity—namely, the preservation of vector addition and scalar
multiplication.
From rotating images in computer graphics to analyzing high-dimensional data in
machine learning, linear transformations are used to simplify complex systems, reveal
hidden patterns, and solve real-world problems efficiently. Their matrix-based
representation makes them both computationally powerful and conceptually elegant.
Understanding linear transformations not only deepens your grasp of linear algebra but
also opens the door to a wide range of applications across science, engineering, and
technology.
