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

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In summary, the linear transformation L: R^3 \rightarrow R^4 has a base for its range given by the set \{u,v\} since the third vector w is a linear combination of the other two vectors, as stated by a theorem about the basis of the range of a transformation. This was determined through the use of the transformation of the basis vectors from the domain of the transformation.
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iNCREDiBLE
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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:
 
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Help anyone??
 
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  • #3
:cry: :cry:
 
  • #4
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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