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

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To determine the basis for the range of the linear transformation L: R^3 → R^4, the vectors L(e_1), L(e_2), and L(e_3) are analyzed. The vectors are identified as u = (2,1,0,1)^T, v = (0,3,3,4)^T, and w = (2,-5,-6,-7)^T. It is established that w can be expressed as a linear combination of u and v, specifically w = u - 2v. Therefore, the basis for the range of L is indeed {u, v}. This conclusion confirms that the range is spanned by the two independent vectors.
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Linear Algebra-question. HELP!

Problem:

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

Determine a base for Range(L).

----

Is the base \{u,v\} since w = u-2v? Is it really that simple? :blushing:
 
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Help anyone??
 
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: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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