Can You Determine the Basis for the Range of a Linear Transformation?

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Homework Help Overview

The discussion revolves around a problem in linear algebra concerning a linear transformation from R^3 to R^4. The original poster presents specific vectors resulting from the transformation and seeks to determine a basis for the range of the transformation.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster questions whether the set \{u,v\} can serve as a basis since one vector appears to be a linear combination of the others. Some participants discuss the implications of a theorem related to the basis of the range and the linear dependence of the vectors provided.

Discussion Status

The discussion includes attempts to clarify the basis for the range and explores the relationship between the vectors. Some participants suggest that the original poster's reasoning about the linear combination is valid, while others affirm the theorem's relevance to the problem.

Contextual Notes

Participants are navigating the definitions and properties of linear transformations and their ranges, with some noting the implications of linear dependence among the vectors provided.

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Linear Algebra-question. HELP!

Problem:

Let [itex]L: R^3 \rightarrow R^4[/itex] be a linear transformation that satisfies:
[itex]L(e_1) = (2,1,0,1)^T = u[/itex]
[itex]L(e_2) = (0,3,3,4)^T = v[/itex]
[itex]L(e_3) = (2,-5,-6,-7)^T = w[/itex].

Determine a base for [itex]Range(L)[/itex].

----

Is the base [itex]\{u,v\}[/itex] since [itex]w = u-2v[/itex]? Is it really that simple? :blushing:
 
Last edited:
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Help anyone??
 
Last edited:
:cry: :cry:
 
yep.

there's a theorem saying that the basis of the range of a transformation is given by the set determined by the transformations of the vectors comprising the basis of the domain of the transformation.

and you used this and noted that the third vector is a linear combination of the other two...

so... yeah, you found the basis for the range of the transformation.
 

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