Is Any Square Matrix a Linear Operator?

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Homework Help Overview

The discussion revolves around the verification of whether any square matrix can be considered a linear operator when viewed as a linear transformation. The subject area pertains to linear algebra and the properties of linear transformations.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the necessary conditions for a square matrix to be a linear operator, with some questioning the need to demonstrate the preservation of sums and scalar multiples as part of the verification process.

Discussion Status

The conversation is ongoing, with participants seeking clarification and guidance on how to approach the problem. Some have suggested specific properties that need to be shown, while others are looking for a more general direction.

Contextual Notes

There appears to be an emphasis on understanding the definition and properties of linear operators, with some participants noting the trivial nature of certain aspects like dimensionality.

jinksys
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Homework Statement



Verify that any square matrix is a linear operator when considered as a linear transformation.

Homework Equations





The Attempt at a Solution



If a square matrix A\inℂ^{n,n} is a linear operator on the vector space C^{n}, where n ≥ 1, then the square matrix A is of the correct dimension to be a linear transform from C->C.
 
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Except for the rather trivial statement about dimension, I see no attempt at this problem- the question is to show that it is linear.
 


Could I get a push in the right direction?
 


show it preserves sums and scalar multiples...
 

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