Question

(A) Prove that over the field C, that Q(i) and Q(2) are isomorphic as vector spaces?

(B) Prove that over the field C, that Q(i) and Q(2) are not isomorphic as fields?

Answer #1

(A) Prove that over the field C, that Q(i) and Q(sqrt(2)) are
isomorphic as vector spaces.
(B) Prove that over the field C, that Q(i) and Q(sqrt(2)) are
not isomorphic as fields

2. Determine whether the following pairs of vector spaces are
isomorphic. If so, give an explicit isomorphism (you must show that
it is indeed an isomorphism); if not, state why not.
a) R5 and P5
b) P8 and Mat33(R)
c) {f ∈ P4 | f(2) = 0} and
{(x1,...,x6) ∈ R6 | x1
= x4 = x5}

Let K be a field. Show that the category of vector spaces over K
is an Abelian category.

Find a basis for each of the following vector spaces and find
its dimension (justify):
(a) Q[ √3 2] over Q
(b) Q[i, √ 5] (that is, Q[i][√ 5]) over Q;

5.1.5. Suppose V1, V2, W are vector spaces over F. Prove that f
: V1 × V2 → W is the zero map if and only if f is both linear and
bilinear.

For each of the following, determine whether it is a vector
space over the given field.
(i) The set of 2 × 2 matrices of real numbers, over R.
(ii) The set of 2 × 2 matrices of real numbers, over C.
(iii) The set of 2 × 2 matrices of real numbers, over Q.

Q 1 Determine whether the following are real vector spaces.
a) The set C with the usual addition of complex numbers and
multiplication by R ⊂ C.
b) The set R2 with the two operations + and · defined
by (x1, y1) + (x2, y2)
= (x1 + x2 + 1, y1 + y2
+ 1), r · (x1, y1) = (rx1,
ry1)

? is a vector field and ? is a scalar field
1. Prove in 3 dimensions: ∇ ∙ (∇ × ? ) = 0
2. Prove in 3 dimensions: ∇ × (∇φ) = 0
3. Prove in 3 dimensions: (?∙∇) ?= −? × (∇ × ?)+ ∇((?∙?)/ 2)

(a) Let a,b,c be elements of a field F. Prove that if a not= 0,
then the equation ax+b=c has a unique solution.
(b) If R is a commutative ring and x1,x2,...,xn are independent
variables over R, prove that R[x σ(1),x σ (2),...,x σ (n)] is
isomorphic to R[x1,x2,...,xn] for any permutation σ of the set
{1,2,...,n}

a)Suppose U is a nonempty subset of the vector space V over
field F. Prove that U is a subspace if and only if cv + w ∈ U for
any c ∈ F and any v, w ∈ U
b)Give an example to show that the union of two subspaces of V
is not necessarily a subspace.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 6 minutes ago

asked 12 minutes ago

asked 12 minutes ago

asked 12 minutes ago

asked 14 minutes ago

asked 21 minutes ago

asked 24 minutes ago

asked 37 minutes ago

asked 47 minutes ago

asked 59 minutes ago

asked 1 hour ago

asked 1 hour ago