Question

**Indicate whether each statement is True or False.
Briefly justify your answers. Please answer all of questions
briefly**

(a) In a vector space, if c⊙⃗u =⃗0, then c= 0.

(b) Suppose that A and B are square matrices and that AB is a non-zero diagonal matrix. Then A is non-singular.

(c) The set of all 3 × 3 matrices A with zero trace (T r(A) = 0) is a vector space under the usual matrix operations of addition and scalar multiplication.

(d) Every non-zero vector space Vhas at least two subspaces. Indicate whether each statement is True or False. Briefly justify your answers.

(e) If x⃗1 and x⃗2 are in the null space of a square matrix A, then any linear combination of x⃗1 andx⃗2 is also in the null space of A.

Answer #1

Answer all of the questions true or false:
1.
a) If one row in an echelon form for an augmented matrix is [0 0 5
0 0]
b) A vector b is a linear combination of the columns of a matrix A
if and only if the
equation Ax=b has at least one solution.
c) The solution set of b is the set of all vectors of the form u =
+ p + vh
where vh is any solution...

Indicate whether each of the following statements is
True or False, and Briefly Justify your answer.
The adjusted R2 is used because the unadjusted
R2 automatically and spuriously increases when extra
explanatory variables are added to the model.

True or False? (Make sure to justify your
answer.)
For this question let ?A be a non-zero ?×?m×n matrix, and ?B an
invertible ?×?n×n matrix.
If {?1,...,??}{x1,...,xk} is a basis for null(?)null(A), then
dim(null(??))=?dim(null(AB))=k.

The trace of a square n×nn×n matrix A=(aij)A=(aij) is the sum
a11+a22+⋯+anna11+a22+⋯+ann of the entries on its main
diagonal.
Let VV be the vector space of all 2×22×2 matrices with real
entries. Let HH be the set of all 2×22×2 matrices with real entries
that have trace 11. Is HH a subspace of the vector space
VV?
Does HH contain the zero vector of VV?
choose H contains the zero vector of V H does not contain the zero
vector...

For each of the following statement, determine it is “always
true” or “always false”. Explain your answer.
(a) For any matrix A, the matrices A and −A have the same row
space. (b) For any matrix A, the matrices A and −A have the same
null space.
(c) For any matrix A, if A has nullity 0, then A is
nonsingular.

(6) Label each of the following statements as True or
False. Provide justification
for your response.
(b) True/False The scalar λ is an eigenvalue of a
square matrix A if and
only if the equation (A − λIn)x = 0 has a nontrivial
solution.
(c) True/False If λ is an eigenvalue of a matrix A, then there is
only
one nonzero vector v with Av = λv.
(d) True/False The eigenspace of an eigenvalue of an n × n matrix...

Indicate whether the following statements are true or
false and justify the answer.
(I) If f and g are functions defined for all real
numbers, and f is an even function, then f o g is also an even
function.
(II) If a function f, defined for all real
numbers, satisfies the the equation f(0) = f(1), then f does not
have an inverse function.

True or false; for each of the statements below, state whether
they are true or false. If false, give an explanation or example
that illustrates why it's false.
(a) The matrix A = [1 0] is not invertible.
[1 -2]
(b) Let B be a matrix. The rowspaces row (B), row (REF(B)) and
row (RREF(B)) are all equivalent.
(c) Let C be a 5 x 7 matrix with nullity 3. The rank of C is
2.
(d) Let D...

_______Indicate whether each statement below is true or false,
and justify your answer
____The Cournot oligopoly model is not useful for analysing
competitionamong firms because (i) firms normally choose prices,
not quantities, and (ii) firmsnormally do not make their decisions
simultaneously

Indicate whether the statement is true or false. if false,
correct it to true:
a) ____dipole- dipole interaction strength increases with
decreasing molecular polarity
b) _____hydrogen bonding occurs only between non polar
hydrogen-containing molecules
c)____ london forces are extra strong dipole-dipole
interactions
d) ___ substances in which hydrogen bonding is present usually
have high vapor pressures.

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