Definition questions for linear algebra

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Discussion Overview

The discussion revolves around foundational concepts in linear algebra, including orthogonal bases, null spaces, column spaces, and projections. Participants seek clarification on definitions and applications relevant to proofs and understanding of the subject.

Discussion Character

  • Homework-related
  • Conceptual clarification

Main Points Raised

  • One participant expresses confusion about the definition and construction of an orthogonal basis, particularly in relation to a set of vectors {v1, v2} in a subspace W.
  • Another participant seeks to understand the concept of the null space of a matrix (or linear transformation) A, specifically for use in proofs.
  • There is a request for clarification on the concept of the column space of a matrix A, with an emphasis on its relevance for proofs.
  • A participant questions the concept of projection, stating they know the formula and application but struggle to visualize or understand its meaning.
  • One participant suggests the Gram-Schmidt process as a method for constructing orthogonal bases.
  • Another participant provides definitions for null space and column space, indicating that the null space consists of vectors that map to the zero vector and that the column space is spanned by the columns of a matrix.
  • Recommendations for linear algebra resources are shared, including a free online textbook and additional notes for easier understanding.

Areas of Agreement / Disagreement

Participants generally express confusion about the same foundational concepts, but there is no consensus on the best approach to understanding or visualizing these concepts. Multiple viewpoints on resources and methods for learning are presented.

Contextual Notes

Some definitions and concepts are presented without detailed explanations, and the discussion reflects varying levels of familiarity with linear algebra, which may affect understanding.

Who May Find This Useful

This discussion may be useful for students struggling with basic linear algebra concepts, particularly those looking for clarification on definitions and seeking recommendations for learning resources.

futeca
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i am having trouble understanding some of the "basic" concepts of my linear algebra...any help would be greatly appreciated

what is an orthogonal basis? and how to construct it? i keep stumbling upon questions asking about construction a orthogonal basis for {v1, v2} in W

what i null A? need to understand the concept fro proofs

what is Col A? also need to understand for proofs

what exactly is a projection? i know the formula for it and i know how to apply it yet i don't understand the concept of what it is or how to picture it

thank you
 
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futeca said:
i am having trouble understanding some of the "basic" concepts of my linear algebra...any help would be greatly appreciated

what is an orthogonal basis? and how to construct it? i keep stumbling upon questions asking about construction a orthogonal basis for {v1, v2} in W

what i null A? need to understand the concept fro proofs

what is Col A? also need to understand for proofs

what exactly is a projection? i know the formula for it and i know how to apply it yet i don't understand the concept of what it is or how to picture it

thank you


Best advice ever: get yourself a good linear algebra book...:)

Ort. basis: read about Gram-Schmidt process and def. of orth. basis

Null A = most probably it means the null (sub)space of a matrix (or linear transformation) A, and it is the set of all vectors that A maps to the zero vector

Col A = probably it means the space spanned by the columns of an n x m matrix in the vector space [itex]\mathbb F^n[/itex]

DonAntonio
 
thank you so much this is great help!

btw do u happen to have any recommendations for a good linear algebra book to buy?
 

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