  (Solution) 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)... | Snapessays.com

(Solution) 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)...

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

map.

c) Prove or disprove: The transpose map T : Mnn(F) -> Mnn(F), which sends a

matrix 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

X and

Y; that is, V = X (internal direct sum) Y . Then every v in V has a unique

decomposition

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:

a) Px

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)

c) PX

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

).

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May 23, 2022

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