Question

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.

Answer #1

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:...

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?

Are the following vector space and why?
1.The set V of all ordered pairs (x, y) with the addition of
R2, 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 M22.

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.

(i) Let u= (u1,u2) and v=
(v1,v2). Show that the following is an inner
product by verifying that the inner product hold
<u,v>= 4u1v1 +
u2v2 +4u2v2
(ii) Let u= (u1,
u2, u3) and v=
(v1,v2,v3). Show that the
following is an inner product by verifying that the inner product
hold
<u,v> =
2u1v1 + u2v2 +
4u3v3

Consider C as a vector space over R in the natural way. Here
vector addition and scalar
multiplication are the usual addition and multiplication of
complex numbers. Show that {1 − i, 1 + i} is
linearly independent. Consider C as a vector space over C in
the natural way. Here vector addition is the
usual addition of complex numbers and the scalar
multiplication is the usual multiplication of a real number
by a complex number. Show that {1 −...

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.

Let S={(0,1),(1,1),(3,-2)} ⊂ R², where R² is a real vector space
with the usual vector addition and scalar multiplication.
(i) Show that S is a spanning set for R²
(ii)Determine whether or not S is a linearly independent set

Let R be a relation on set RxR of ordered pairs of real numbers
such that (a,b)R(c,d) if a+d=b+c. Prove that R is an equivalence
relation and find equivalence class [(0,b)]R

Let
R4
have the inner product
<u, v> =
u1v1 +
2u2v2 +
3u3v3 +
4u4v4
(a)
Let w = (0, 6,
4, 1). Find ||w||.
(b)
Let W be the
subspace spanned by the vectors
u1 = (0, 0, 2,
1), and u2 = (3, 0, −2,
1).
Use the Gram-Schmidt process to transform the basis
{u1,
u2} into an
orthonormal basis {v1,
v2}. Enter the
components of the vector v2 into the
answer box below, separated with commas.

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