In Exercises 9-16, find a basis for the eigenspace...

In Exercises $9-16,$ find a basis for the eigenspace corresponding to each listed eigenvalue.
$$
A=\left[\begin{array}{rrr}{4} & {2} & {3} \\ {-1} & {1} & {-3} \\ {2} & {4} & {9}\end{array}\right], \lambda=3
$$

Key Concepts

Solving Linear Systems

Determining the eigenspace typically involves solving the homogeneous linear system (A - ?I)x = 0. Techniques such as row reduction or Gaussian elimination are used to solve these systems, enabling the identification of all vectors that satisfy the eigenvalue equation.

Basis

A basis of a vector space is a set of linearly independent vectors that span the entire space. In the context of eigenspaces, finding a basis means identifying the smallest set of eigenvectors from which every other eigenvector in the space can be linearly composed, providing a complete description of the subspace.

Eigenvalue and Eigenvector

Eigenvalues are scalars associated with a matrix such that there exists a nonzero vector (the eigenvector) which, when the matrix is applied to it, results only in a scaling of the vector. This concept is central in understanding linear transformations, as it identifies directions that remain invariant aside from a scaling factor.

Eigenspace

The eigenspace corresponding to a given eigenvalue is the set of all eigenvectors associated with that eigenvalue, along with the zero vector. It forms a subspace of the vector space and can be found by determining the null space of the matrix obtained by subtracting the eigenvalue times the identity matrix from the original matrix.

Null Space (Kernel) of a Matrix

The null space of a matrix consists of all vectors that are mapped to the zero vector by the matrix transformation. When analyzing eigenvalues, the null space of (A - ?I) represents all eigenvectors corresponding to the eigenvalue ?, highlighting the inherent link between eigenvalues and the structure of the solution space of homogeneous systems.

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