The questions are attached.1a) Prove or disprove: The trace function tr: Mnn(F) -> F is a linear map.
b) Prove or disprove: The determinant function det: Mnn(F) -> F is a linear
c) Prove or disprove: The transpose map T : Mnn(F) -> Mnn(F), which sends a
to its transpose A^t, is a linear map.
2) Suppose that the vector space V is the (internal) direct sum of the subspaces
Y; that is, V = X (internal direct sum) Y . Then every v in V has a unique
v = x + y where x in X and y in Y . Hence we can define maps PX: V -> X and
PY: V -> Y
by the equations Px(v) = x and Py(v) = y. Prove the following:
Px = Px and Py
Py = Py
b) Px + Py = Iv , where Iv : V -> V is the identity map (that is, Iv(v) = v for
all v in V)
PY = OV and PY
PX = Ov , where Ov : V -> V is the zero map (that is,
Ov(v) = 0 for all v in V)
3) Let W be a subspace of the vector space V , and let V = W be the quotient
space. Define the
function pi : V -> V / W by the equation pi(v) = v + W for all v in V .
a) Prove that pi is a linear map.
b) Prove that pi is onto (for every z in V / W we have z = pi(v) for some v in V
This question was answered on: May 23, 2022
This attachment is locked
Our expert Writers have done this assignment before, you can reorder for a fresh, original and plagiarism-free copy and it will be redone much faster (Deadline assured. Flexible pricing. TurnItIn Report provided)
May 23, 2022EXPERT
We have top-notch tutors who can do your essay/homework for you at a reasonable cost and then you can simply use that essay as a template to build your own arguments.
You can also use these solutions: