Let
M 2,2 be the set of all 2x2 matrices determine whether the
following subspaces.
A) the set of all 2x2 diagnol matrices
B) the set of all matrices with a12 entry
C) the set of all 2x2 triangular matrices
Determine if the following subsets are subspaces:
1. The set of grade 7 polynomials
2. The set of polynomials of degree 5 such that P (0) = 0
3. The set of continuous functions such that f (0) = 2
please provide explanations within your proof
1) Prove that if ?1, ?2 are subspaces of ? such that ?1 + ?2 =
?1 ⊕ ?2 then there is a linear isomorphism
? : ?1 × ?2 → ?1 ⊕ ?2.
(Recall: Given two sets ?1, ?2 the cross product ?1 × ?2 is
the set of elements (?, ?)
where ? ∈ ?1 and ? ∈ ?2 with pointwise addition and scalar
multiplication.)
2) Let ? be a finite dimensional...
Use Theorem 4.2.1 to determine which of the following sets of
vectors are subspaces of F(−∞,∞). (Justify!)
(a) The set of all functions f ∈ F(−∞,∞) for which f(0) = 0.
(b) The set of all functions f ∈ F(−∞,∞) for which f(1) = 0.
(c) The set of all functions f ∈ F(−∞,∞) for which f(0) = 1.
(d) The set of all functions f ∈ F(−∞,∞) for which f(−x) = f(x)
for all x ∈ R
Determine the [OH−] of a solution that is 0.150 M in CO32−.
Express your answer using two significant figures.
[OH] =
M
Part B
Determine the pH of a solution that is 0.150 M in CO32−.
Express your answer to two decimal places.
pH =
Part C
Determine the pOH of a solution that is 0.150 M in CO32−.
Express your answer to two decimal places.
pOH =