Highway said:Homework Statement
I have a matrix and need to show that it is a subspace of ℝn using the eigenspace identity of: Ax = λx, where x is the eigenvector.
Homework Equations
The Attempt at a Solution
I know that for a subspace, you need to show that it holds under addition, scalar multiplication, and that the zero vector is present. But I don't know how to get started on this... any hints? Thanks!
"it" = what? Math students should never use the word "it" unless the antecedent is obvious to the most casual observer.Highway said:I know that for a subspace, you need to show that it holds under addition, scalar multiplication, and that the zero vector is present. But I don't know how to get started on this... any hints? Thanks!
Mark44 said:I don't believe that your problem description is an accurate depiction of what you're supposed to do. Here's what I think that problem actually is:
For a given n x n matrix A, with a given eigenvalue λ, show that the solutions of the equation Ax = λx form a subspace of Rn. In other words, show that the eigenvectors associated with the eigenvalue λ form a subspace of Rn.
Mark44 said:"it" = what? Math students should never use the word "it" unless the antecedent is obvious to the most casual observer.
Mark44 said:See? Even you're confused. This really has almost nothing to do with any set of n x n matrices - just one particular matrix, and the set of vectors that are the eigenvectors associated with one eigenvalue of that matrix.
Your sentence with "it" removed is:
I know that to show that this set of vectors is a subspace, you need to show that [STRIKE]it holds under addition[/STRIKE] this set is closed under vector addition and scalar multiplication, and that the zero vector is present.
So you need to check 3 things.
1) Is the 0 vector in this set? I.e., is A0 = λ0 a true statement.
2) Is the set closed under vector addition?
3) Is the set closed under scalar multiplication?
In linear algebra, a subspace is a subset of a vector space that satisfies the properties of a vector space. This means that it is closed under addition and scalar multiplication, and contains the zero vector.
To prove that a set is a subspace, you must show that it satisfies the three properties of a vector space: closure under addition, closure under scalar multiplication, and containing the zero vector. This can be done by showing that any two vectors in the set can be added to give another vector in the set, and that any scalar multiple of a vector in the set is also in the set.
A subspace is a subset of a vector space that satisfies the properties of a vector space, while a span is the set of all possible linear combinations of a given set of vectors. A subspace must contain the zero vector and be closed under addition and scalar multiplication, while a span does not necessarily have to satisfy these properties.
No, a subspace must contain at least two vectors. This is because a subspace must be closed under addition, and if it only contains one vector, there is no other vector that can be added to it to give another vector in the subspace.
To prove that a set of matrices is a subspace, you must show that it satisfies the three properties of a vector space: closure under addition, closure under scalar multiplication, and containing the zero vector. This can be done by showing that any two matrices in the set can be added to give another matrix in the set, and that any scalar multiple of a matrix in the set is also in the set.