Question

Are the following vector space and why?

1.The set V of all ordered pairs (x, y) with the addition of
R^{2}, but scalar multiplication a(x, y) = (x, y) for all a
in R.

2. The set V of all 2 × 2 matrices whose entries sum to 0;
operations of M_{22}.

Answer #1

hi if you have any doubts please comment here I will help you

Let V be the set of all ordered pairs of real numbers. Consider
the following addition and scalar multiplication operations V. Let
u = (u1, u2) and v = (v1, v2).
• u ⊕ v = (u1 + v1 + 1, u2 + v2 + )
• ku = (ku1 + k − 1, ku2 + k − 1)
Show that V is not a vector space.

Let V be the set of all ordered pairs of real numbers. Consider
the following addition and scalar multiplication operations V. Let
u = (u1, u2) and v = (v1, v2).
• u ⊕ v = (u1 + v1 + 1, u2 + v2 + )
• ku = (ku1 + k − 1, ku2 + k − 1)
1)Show that the zero vector is 0 = (−1, −1).
2)Find the additive inverse −u for u = (u1, u2). Note:...

Exercise 9.1.11 Consider the set of all vectors in R2,(x, y)
such that x + y ≥ 0. Let the vector space operations be the usual
ones. Is this a vector space? Is it a subspace of R2?
Exercise 9.1.12 Consider the vectors in R2,(x, y) such that xy =
0. Is this a subspace of R2? Is it a vector space? The addition and
scalar multiplication are the usual operations.

Verify this axiom of a vector space.
Vector space:
A subspace of R2: the set of all dimension-2 vectors
[x; y] whose entries x and y are odd integers.
Axiom 1:
The sum u + v is in V.

Consider the set of all ordered pairs of real numbers with
standard vector addition but with scalar multiplication defined
by k(x,y)=(k^2x,k^2y).
I know this violates (alpha + beta)x = alphax + betax, but I'm
not for sure how to figure that out? How would I figure out which
axioms it violates?

Show that the set Vof all 3 x 3 matrices with distinct entries
also combination of positive and negative numbers is a vector space
if vector addition is defined to be matrix addition and vector
scalar multiplication is defined to be matrix scalar
multiplication

Let V be the set of all triples (r,s,t) of real numbers with the
standard vector addition, and with scalar multiplication in V
deﬁned by k(r,s,t) = (kr,ks,t). Show that V is not a vector space,
by considering an axiom that involves scalar multiplication. If
your argument involves showing that a certain axiom does not hold,
support your argument by giving an example that involves speciﬁc
numbers. Your answer must be well-written.

Consider the set V = (x,y) x,y ∈ R with the following two
operations: • Addition: (x1,y1)+(x2,y2)=(x1 +x2 +1, y1 +y2 +1) •
Scalarmultiplication:a(x,y)=(ax+a−1, ay+a−1). Prove or disprove:
With these operations, V is a vector space over R

Prove that the set V of all polynomials of degree ≤ n including
the zero polynomial is vector space over the field R under usual
polynomial addition and scalar multiplication. Further, find the
basis for the space of polynomial p(x) of degree ≤ 3. Find a basis
for the subspace with p(1) = 0.

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...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 9 minutes ago

asked 12 minutes ago

asked 37 minutes ago

asked 40 minutes ago

asked 43 minutes ago

asked 50 minutes ago

asked 51 minutes ago

asked 51 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago