From A First Course in Linear Algebra
Version 2.20
© 2004.
Licensed under the GNU Free Documentation License.
http://linear.ups.edu/
Summary Domain and codomain are polynomials. Domain has dimension 5, while codomain has dimension 6. Is injective, can’t be surjective.
A linear transformation: (Definition LT)
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T : {P}_{4} → {P}_{5},\quad T\left (p(x)\right ) = (x − 2)p(x)
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A basis for the null space of the linear transformation: (Definition KLT)
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\left \{\ \right \}
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Injective: Yes. (Definition ILT)
Since the kernel is trivial Theorem KILT tells us that the linear transformation is
injective.
A basis for the range of the linear transformation: (Definition RLT)
Evaluate the linear transformation on a standard basis to get a spanning set for
the range (Theorem SSRLT):
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\left \{x − 2,\kern 1.95872pt {x}^{2} − 2x,\kern 1.95872pt {x}^{3} − 2{x}^{2},\kern 1.95872pt {x}^{4} − 2{x}^{3},{x}^{5} − 2{x}^{4},{x}^{6} − 2{x}^{5}\right \}
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If the linear transformation is injective, then the set above is guaranteed to be linearly independent (Theorem ILTLI). This spanning set may be converted to a “nice” basis, by making the vectors the rows of a matrix (perhaps after using a vector reperesentation), row-reducing, and retaining the nonzero rows (Theorem BRS), and perhaps un-coordinatizing. A basis for the range is:
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\left \{−{1\over
32}{x}^{5} + 1,\kern 1.95872pt − {1\over
16}{x}^{5} + x,\kern 1.95872pt −{1\over
8}{x}^{5} + {x}^{2},\kern 1.95872pt −{1\over
4}{x}^{5} + {x}^{3},\kern 1.95872pt −{1\over
2}{x}^{5} + {x}^{4}\right \}
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Surjective: No. (Definition SLT)
The dimension of the range is 5, and the codomain
({P}_{5}) has
dimension 6. So the transformation is not surjective. Notice too that since the
domain {P}_{4}
has dimension 5, it is impossible for the range to have a dimension greater than 5,
and no matter what the actual definition of the function, it cannot possibly be
surjective in this situation.
To be more precise, verify that 1 + x + {x}^{2} + {x}^{3} + {x}^{4}∉ℛ\kern -1.95872pt \left (T\right ), by setting the output equal to this vector and seeing that the resulting system of linear equations has no solution, i.e. is inconsistent. So the preimage, {T}^{−1}\left (1 + x + {x}^{2} + {x}^{3} + {x}^{4}\right ), is nonempty. This alone is sufficient to see that the linear transformation is not onto.
Subspace dimensions associated with the linear transformation. Examine parallels with earlier results for matrices. Verify Theorem RPNDD.
Invertible: No.
The relative dimensions of the domain and codomain prohibit any possibility of
being surjective, so apply Theorem ILTIS.
Matrix representation (Definition MR):