Spanning sets, eigenvalues, eigenvectors etc .

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Discussion Overview

The discussion revolves around the concepts of spanning sets, eigenvalues, and eigenvectors, with a focus on their definitions, importance, and applications in linear algebra. Participants explore theoretical aspects and practical steps related to these topics, particularly in the context of preparing for an exam.

Discussion Character

  • Exploratory
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant seeks clarification on the definition of a spanning set and its significance, particularly in relation to preparing for an exam.
  • Another participant explains that a set of vectors is a spanning set for a vector space if any element of that space can be expressed as a linear combination of the vectors in the set, assuming the space is finite-dimensional.
  • The explanation includes details about eigenvalues and eigenvectors, noting their role in encoding geometrical information about linear maps, and mentions specific cases such as singular matrices and diagonalization.
  • There is a mention of the importance of eigenvalues and eigenvectors in various mathematical contexts, including operator theory and applications in physics.
  • A participant asks for steps to find a spanning set for a specific space defined by a matrix equation.
  • Another participant responds by stating that this space is equivalent to the nullspace of a related matrix equation.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and seek clarification on specific concepts, indicating that there is no consensus on the definitions and implications of spanning sets, eigenvalues, and eigenvectors. The discussion includes both exploratory questions and technical explanations, reflecting differing perspectives and levels of familiarity with the topics.

Contextual Notes

Some participants may have assumptions about the definitions and properties of linear algebra concepts that are not explicitly stated. The discussion includes unresolved mathematical steps related to finding spanning sets and the implications of eigenvalues and eigenvectors.

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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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.
 


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?
 


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

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