Question

For X,Y ∈ R^(n×n). There exists A ∈ R^(n×n) such that XA = Y if and only if the column space of Y is a subspace of the column space of X. Is this statement true or not, prove your answer.

Answer #1

Exercise1.2.1: Prove that if t > 0 (t∈R),
then there exists an n∈N such that 1/n^2 < t.
Exercise1.2.2: Prove that if t ≥ 0(t∈R), then
there exists an n∈N such that n−1≤ t < n.
Exercise1.2.8: Show that for any two real
numbers x and y such that x < y, there exists an irrational
number s such that x < s < y. Hint: Apply the density of Q to
x/(√2) and y/(√2).

Prove: Let x,y be in R such that x < y.
There exists a z in R such that x < z <
y.
Given:
Axiom 8.1. For all x,y,z in
R:
(i) x + y = y + x
(ii) (x + y) + z = x + (y + z)
(iii) x*(y + z) = x*y + x*z
(iv) x*y = y*x
(v) (x*y)*z = x*(y*z)
Axiom 8.2. There exists a real number 0 such that
for all...

Determine all values of n for which the following statement is
true: There exists integers x and y such that 63x + 147y = n.
Give a convincing argument to justify your answer.

1. For n exists in R, we define the function f by f(x)=x^n, x
exists in (0,1), and f(x):=0 otherwise. For what value of n is f
integrable?
2. For m exists in R, we define the function g by g(x)=x^m, x
exists in (1,infinite), and g(x):=0 otherwise. For what value of m
is g integrable?

1)T F: All (x, y, z) ∈ R 3 with x = y + z is a subspace of R 3
9
2) T F: All (x, y, z) ∈ R 3 with x + z = 2018 is a subspace of R
3
3) T F: All 2 × 2 symmetric matrices is a subspace of M22. (Here
M22 is the vector space of all 2 × 2 matrices.)
4) T F: All polynomials of degree exactly 3 is...

A function f”R n × R m → R is bilinear if for all x, y ∈ R n and
all w, z ∈ R m, and all a ∈ R: • f(x + ay, z) = f(x, z) + af(y, z)
• f(x, w + az) = f(x, w) + af(x, z) (a) Prove that if f is
bilinear, then (0.1) lim (h,k)→(0,0) |f(h, k)| |(h, k)| = 0. (b)
Prove that Df(a, b) · (h, k) = f(a,...

Using the following axioms:
a.) (x+y)+x = x +(y+x) for all x, y in R (associative law of
addition)
b.) x + y = y + x for all x, y elements of R (commutative law of
addition)
c.) There exists an additive identity 0 element of R (x+0 = x
for all x elements of R)
d.) Each x element of R has an additive inverse (an inverse with
respect to addition)
Prove the following theorems:
1.) The additive...

(a) Let the statement,
∀x∈R,∃y∈R G(x,y), be true for predicate G(x,y).
For each of the following statements, decide if the statement is
certainly true, certainly false,or possibly true, and justify your
solution.
1
(i)
G(3,4)
(ii)
∀x∈RG(x,3)
(iii)
∃y G(3,y)
(iv)
∀y¬G(3,y)(v)∃x G(x,4)

Consider the relation R defined on the set R as follows: ∀x, y ∈
R, (x, y) ∈ R if and only if x + 2 > y.
For example, (4, 3) is in R because 4 + 2 = 6, which is greater
than 3.
(a) Is the relation reflexive? Prove or disprove.
(b) Is the relation symmetric? Prove or disprove.
(c) Is the relation transitive? Prove or disprove.
(d) Is it an equivalence relation? Explain.

Consider the following subset:
W =(x, y, z) ∈ R^3; z = 2x - y from R^3.
Of the following statements, only one is true. Which?
(1) W is not a subspace of R^3
(2) W is a subspace of R^3
and {(1, 0, 2), (0, 1, −1)} is a base of W
(3) W is a subspace of R^3
and {(1, 0, 2), (1, 1, −3)} is a base of W
(4) W is a subspace of R^3
and...

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