Linear Algebra: UNIT II: Linear Transformations and Diagonalization

Linear Transformations: Projection

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Projection Definition for Engineering Maths or Mathematics - Linear Transformations: Projection

Projection

Definition

Let V be a vector space and W₁ and W₂ be subspaces of V such that V=W₁ W₂. The function T: V→ V defined by T (x)=x₁ where x=x₁+x² with x₁ ∈ W₁ and x² ∈ W₂ is called the projection of V on W₁ or the projection on W₁ along W₂.

Definition

Let V be a vector space and let T: V→V be linear. A subspace W of V is said to be T−invariant if T (x) ∈ W for every x ∈ W, (ie) T (W) W. If W is T−invariant, we define the restriction of T on W to be the function T_w: W → W defined by

T_w(x)=T(x) for all x ∈ W.

Linear Algebra: UNIT II: Linear Transformations and Diagonalization : Tag: maths, mathematics : - Linear Transformations: Projection

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Linear Algebra: UNIT II: Linear Transformations and Diagonalization

Linear Transformation and Diagonalization

Definition and Properties of Linear Transformation

Linear Transformation: Example Solved Problems

Null space (or) Kernel N(T) and Range (or) Image R(T)

Null space (or) Kernel N(T) and Range (or) Image R(T): Example Solved Problems

Null space (or) Kernel N(T) and Range (or) Image R(T): Theorem Part 2

Null space (or) Kernel N(T) and Range (or) Image R(T): Example Solved Problems - Part 2

Linear Transformations: Projection

Linear Transformation: Example Solved Problems

Matrix Representation of a Linear Transformation

Matrix Representation of a Linear Transformation: Example Solved Problems

Matrix Representation of a Linear Transformation: Theorems

Matrix Representation of a Linear Transformation: Example Solved Problems Using Theorems

Definition of Diagonalizable, Eigenvalues and Eigenvector

Eigenvalues and Eigenvectors: Theorem 1

Eigenvalues and Eigenvectors: Theorem 1 - Example Solved Problems

Eigenvalues and Eigenvectors: Theorem 2

Eigenvalues and Eigenvectors: Theorem 2 - Example Solved Problems

Linear Transformations: Characteristic polynomial

Characteristic Polynomial: Example Solved Problems

Eigenvalues and Eigenvectors: Theorem 3

Eigenvalues and Eigenvectors: Theorem 3 - Example Solved Problems

Diagonalizability: Theorems Part 1

Diagonalizability: Theorems Part 1 - Example Solved Problems

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