Nilai dan Vektor Eigen part 1

Summary

This lecture explores advanced algebra concepts, focusing on eigenvalues and eigenvectors. It defines the relationship between linear operators and eigenvectors, demonstrating how these vectors are scaled by a constant factor, the eigenvalue. Through practical examples, including matrix transformations and differentiation operators, the session illustrates how to identify eigenvectors and calculate corresponding eigenvalues. Theorems like Theorem 342 further clarify the structural properties of eigenvectors, showing how they form subspaces in vector spaces. The content is foundational for understanding linear algebra and its applications in various mathematical fields.

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Q & A

What is the definition of eigenvector and eigenvalue in linear algebra?

An eigenvector is a non-zero vector that, when mapped by a linear operator, results in a scalar multiple of itself. The corresponding scalar is called the eigenvalue.

What does the term 'linear operator' refer to in the context of this lecture?

A linear operator is a linear map or transformation that operates on a vector space, often represented by a function that maps vectors to vectors within the same or another vector space.

In the example provided, why is the vector [1, 1] an eigenvector of the linear operator?

The vector [1, 1] is an eigenvector because when mapped by the operator, it results in a scalar multiple of itself, specifically a multiple of 3.

Can there be multiple eigenvectors for the same eigenvalue? Give an example.

Yes, there can be multiple eigenvectors for the same eigenvalue. For instance, both [1, 1] and [2, 2] are eigenvectors with the eigenvalue 3 in the given example.

What is the significance of the example with the function e^x?

The example with e^x illustrates how a function can be an eigenvector of an operator, where the second derivative of e^x is proportional to e^x itself, making it an eigenvector with eigenvalue 1.

Why is the function x^2 not considered an eigenvector in the provided example?

The function x^2 is not considered an eigenvector because its second derivative does not result in a scalar multiple of x^2, which violates the condition for being an eigenvector.

What is the theorem discussed in section 3.4 about eigenvectors and eigenvalues?

The theorem states that a vector is an eigenvector associated with an eigenvalue if and only if it is in the kernel of the linear operator. Additionally, the set of all eigenvectors corresponding to an eigenvalue, along with the zero vector, forms a subspace.

What does it mean for a set of eigenvectors to form a subspace?

A set of eigenvectors forms a subspace if it is non-empty, closed under vector addition, and closed under scalar multiplication. This ensures that the set is a valid subspace of the vector space.

How is the kernel of a linear operator defined?

The kernel of a linear operator consists of all vectors that map to the zero vector under the transformation defined by the operator.

What does the term 'super-space' refer to in the theorem, and how is it proven?

A super-space (or subspace) is the set of all eigenvectors corresponding to a particular eigenvalue, along with the zero vector. The proof shows that this set is closed under vector addition and scalar multiplication, making it a subspace.