Linear Algebra: UNIT II: Linear Transformations and Diagonalization

Eigenvalues and Eigenvectors: Theorem 1 - Example Solved Problems

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Important Example Solved Problems - Engineering Maths or Mathematics - Eigenvalues and Eigenvectors: Theorem 1 - Example Solved Problems

EIGENVALUES AND EIGENVECTORS – THEOREM 1

EXAMPLE PROBLEMS

Example 1

Let A = . Find L_A and the basis β.

Solution:

L_A(v₁) = = ‒2v₁

v₁ is an eigenvector of L_A and hence of A.

Here λ₁= −2 is the eigenvalue corresponding to v₁.

Furthermore,

L_A = (v₂)

= = 5v2

Here λ₂ = 5 is the eigenvalue corresponding to v₂.

Β = {} is a basis for R² consisting of eigenvector of A.

Q = and D = Q^‒1AQ = [L_A]_β

[L_A]_β = which is diagonal matrix.

Linear Algebra: UNIT II: Linear Transformations and Diagonalization : Tag: maths, mathematics : - Eigenvalues and Eigenvectors: Theorem 1 - Example Solved Problems

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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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