Question

Prove why the following statement is true:

A determinant is linear as a function of each of its vector
arguments.

Answer #1

What is the definition of a linear function and is the
determinant function a linear function? Please describe how to
know.

2. Please justify and prove each statement
a) Prove that a finite positive linear combination of metrics is
a metric. If it is infinite, will it be metric?
b) Is the difference of two metrics a metric?

2. Please justify and prove each statement
a) Prove that a finite positive linear combination of metrics is
a metric. If it is infinite, will it be metric?
b) Is the difference between two metrics a metric?

For each of the following linear operators T on vector space V,
compute the determinant T and the characteristic polynomial of
T.
(a). V = R2 , T(a, b) = (2a - b, 5a + 3b)
(b). V = R3 , T(a, b, c) = (a - 3b + 2c, -2a + b + c,
4a - c)
(c). V = P3(R) , T(a, b, c) = T(a + bx +
cx2 + dx3) = (a - c) + (-a...

Which of the following are true in simple linear regression?
True or False for each
There is only one independent variable (X).
Y is the dependent variable.
The relationship between X and Y is described by a linear
function.
Changes in Y are assumed to be related to changes in X.
X is the independent variable because Y is dependent on X

1. Decide whether each statement is true or false. Prove your
answer (i.e. prove that it is true or prove that it is false.)
(a) There exists a nonzero integer α such that α · β is an
integer for every rational number β.
(b) For every rational number β, there exists a nonzero integer
α such that α · β is an integer.

Prove that the composition of two linear fractional functions
is also a linear fractional function

For each of the following statements: if the statement is true,
then give a proof; if the
statement is false, then write out the negation and prove that.
For all sets A;B and C, if B n A = C n A, then B = C.

Prove the statement true or use a counter-example to explain why
it is false.
Let a, b, and c be natural numbers. If (a*c) does not divide
(b*c), then a does not divide b.

For each statement below, either show that the statement is true
or give an example showing that it is false. Assume throughout that
A and B are square matrices, unless otherwise specified.
(a) If AB = 0 and A ̸= 0, then B = 0.
(b) If x is a vector of unknowns, b is a constant column vector,
and Ax = b has no solution, then Ax = 0 has no solution.
(c) If x is a vector of...

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