Consider linear transformations T from V to W and L from W to U. If ker T and ker L are both finite dimensional, show that ker ( L ∘ T ) is finite dimensional as well, and dim ( ker ( L ∘ T ) ) ≤ dim ( ker T ) + dim ( ker L ) .Hint. Restrict T to ker ( L ∘ T ) and apply the rank-nullity theorem, as presented in Exercise 82.
Consider linear transformations T from V to W and L from W to U. If ker T and ker L are both finite dimensional, show that ker ( L ∘ T ) is finite dimensional as well, and dim ( ker ( L ∘ T ) ) ≤ dim ( ker T ) + dim ( ker L ) .Hint. Restrict T to ker ( L ∘ T ) and apply the rank-nullity theorem, as presented in Exercise 82.
Solution Summary: The author explains that if T is a linear transformation from V to W, and mathrmkerTand
Consider linear transformations T from V to W and L from W to U. If ker T and ker L are both finite dimensional, show that
ker
(
L
∘
T
)
is finite dimensional as well, and
dim
(
ker
(
L
∘
T
)
)
≤
dim
(
ker
T
)
+
dim
(
ker
L
)
.Hint. Restrict T to
ker
(
L
∘
T
)
and apply the rank-nullity theorem, as presented in Exercise 82.
The mapping L(x) = ax + b is linear.
إختر واحدا
an
İhi O
Let T: R? → P2(R) and U : Rª → M2x2 (R) be linear transformations.
A student claims U must be invertible because dim(Rª) = dim(M2x2 (R)). If the student is correct, prove
their claim. If the student is not correct, explain why and give an example to illustrate. Clearly state
whether or not the student is correct as part of your solution.
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