(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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This question was answered on: May 23, 2022

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