Linear_Transformation_Presentation_1.pptx

Linear_Transformation_Presentation.pptx

1. Group Members Introduction Group Members: 1. [Your Name] 2. [Member 2] 3. [Member 3] 4. [Member 4]

2. Project Title Linear Transformation

3. What is a Linear Transformation? A linear transformation is a function that maps vectors while preserving: 1. Additivity: T(u+v) = T(u) + T(v) 2. Homogeneity: T(cu) = cT(u)

4. Example of Linear Transformation T(x, y) = (2x + y, x - 3y) Checking: 1. Additivity holds 2. Homogeneity holds => It is a valid linear transformation.

5. Matrix Representation Any linear transformation can be written as: T(x, y) = A * [x, y] For example, A = [2 1; 1 -3]

6. Geometric Interpretation Scaling, Rotation, Reflection, and Projection are common transformations. Each transformation has a specific matrix representation.

7. Example 1 - Scaling Scaling transformation matrix: A = [k 0; 0 k] Effect: Enlarges or shrinks objects.

8. Example 2 - Rotation Rotation transformation matrix: A = [cosθ -sinθ; sinθ cosθ] Effect: Rotates vectors by θ degrees.

9. Example 3 - Reflection Reflection across x-axis: A = [1 0; 0 -1] Effect: Mirrors objects over x-axis.

10. Kernel and Range Kernel (Null Space): Ker(T) = {x | T(x) = 0} Range (Image): Im(T) = {T(x) | x ∈ V}

11. Rank-Nullity Theorem States that: dim(Ker(T)) + dim(Im(T)) = dim(V) Helps analyze transformations.

12. Applications 1. Computer Graphics 2. Machine Learning 3. Signal Processing 4. Robotics 5. Cryptography

13. Example Problem Find the transformation matrix for T(x, y) = (3x - 2y, 4x + 5y).

14. Solution T(x, y) = A * [x; y] A = [3 -2; 4 5] Thus, the transformation matrix is A = [3 -2; 4 5].

15. Summary Linear Transformations: ✔ Preserve vector addition & scalar multiplication ✔ Represented as matrices ✔ Used in real-world applications.

16. Thank You Any questions? 😊