Spanning sets, eigenvalues, eigenvectors etc .

The nullspace of a matrix can be found by row reducing the matrix and finding the special solutions. In summary, a spanning set is a set of vectors that can be used to write any element in a vector space as a linear combination of those vectors. Eigenvalues and eigenvectors are important because they provide geometric information about linear maps and can make computations easier. To find a spanning set for the space AX=A^TX, one can find the nullspace of the matrix (A-A^T) by row reducing and finding special solutions. This method can be applied to a 5x5 matrix.
  • #1
phy
spanning sets, eigenvalues, eigenvectors etc...

can anyone please explain to me what a spanning set is? I've been having some difficulty with this for a long time and my final exam is almost here.
also, what are eigenvalues and eigenvectors? i know how to calculate them but i don't understand why they are so important. thanks.
 
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  • #2
A set of vectors S is a spanning set for a vector space V if any element of V can be written as a (finite) sum of linear combinations of S (We'll assume V's finite dimensional)
That is given v in V there exists vectors s_1,s_2...,s_r in S and elements t_1...t_r in the underlying field such that [tex]v = t_1s_1+\ldots+t_rs_r[/tex] note that the order is you give me v in V, and then I pick these elements depending on the v you've given me.

As for eigenvalues, and eigenvectors, well, they encode a lot of geometrical information about the linear map. For instance if any evalue is zero the matrix is singular. If there are n distinct eigenvalues of an nxn matrix then there is a basis that diagonalizes it (if there are fewer it may or may not diagonalize, you need to know the minimal polynomial as well as the characteristic one). If we may switch emphasis, an eigenvector spans an invariant subspace. Lots of maths uses the idea of invariants. DIfferentiation is a linear map, its eigenvectors are the functions e^{kx}. Expressing things in terms of eigenvectors makes computation easier: if v= v_1+...v_m is a decomposition into eigenvectors then you can work out the image of v easily.
Then there is the fact that it might not be the eigenvalues/vectors that are the important thing but results about eigenvalues and vectors that count. If H is an Hermitian matrix it is diagonalizable, then there's Sylvester's law, the relations with determinants and traces (traces being very important in physics), generalizing these ideas leads to interesting results in operator theory and such as C* algebras that seem to be the way to think about quantum gravity and the like.
 
  • #3


what are the steps that need to be followed to find a spanning set for the space AX=A^TX where we are given a 5x5 matrix?
 
  • #4


It is the same as the space (A-A^T)X = 0, i.e. the nullspace of A-A^T.
 
  • #5


A spanning set is a set of vectors that can be combined in different ways to create any other vector in a given vector space. In other words, the spanning set "spans" the entire space. This is important because it allows us to describe and manipulate vectors in a more compact way, instead of having to specify each vector individually.

Eigenvalues and eigenvectors are important concepts in linear algebra. An eigenvector is a vector that, when multiplied by a given matrix, results in a scalar multiple of itself. The corresponding scalar multiple is called the eigenvalue. In other words, an eigenvector remains in the same direction but may change in magnitude when multiplied by a matrix.

Eigenvalues and eigenvectors are important because they provide insight into the behavior of a matrix. They can be used to simplify calculations, identify special properties of a matrix, and solve systems of linear equations. They also have applications in fields such as physics, engineering, and computer science.

It is important to understand the concepts of spanning sets, eigenvalues, and eigenvectors in order to fully grasp the power and applications of linear algebra. I recommend practicing with different examples and seeking help from your professor or a tutor if you are still struggling. Best of luck on your exam!
 

1. What is a spanning set?

A spanning set is a set of vectors that can be used to generate all other vectors in a vector space through linear combinations. In other words, a spanning set contains enough information to represent all possible vectors in a given vector space.

2. Why is it important to find a spanning set?

Finding a spanning set is important because it allows us to understand the structure and properties of a vector space. It also helps us to solve problems involving linear transformations and to determine whether a set of vectors is linearly independent.

3. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are important concepts in linear algebra that are used to understand the behavior of linear transformations. An eigenvector is a non-zero vector that, when multiplied by a matrix, results in a scalar multiple of the original vector. The corresponding scalar multiple is known as the eigenvalue.

4. How are eigenvalues and eigenvectors used in applications?

Eigenvalues and eigenvectors have many practical applications, including image and signal processing, data compression, and solving systems of differential equations. They are also used in quantum mechanics and in analyzing the stability of systems in engineering.

5. Can a matrix have complex eigenvalues and eigenvectors?

Yes, a matrix can have complex eigenvalues and eigenvectors. In fact, complex eigenvalues and eigenvectors are often used in applications such as quantum mechanics and electrical engineering. Complex eigenvalues and eigenvectors can also provide more accurate solutions to certain problems compared to real eigenvalues and eigenvectors.

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