Question

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;

Answer #1

Find the dimension of each of the following vector spaces.
a.) The space of all n x n upper triangular matrices A with
zeros in the main diagonal.
b.) The space of all n x n symmetric matrices A.
c.) The space of all n x n matrices A with zeros in the first
and last columns.

Give an example of 4 vector spaces of dimension 3.

Consider the vector u1=(2,0,2), u2=(4,1,-1), u3=( 0,1,-5),
u4=(3,0,2)
a) Find the dimension and a basis for U= span{ u1,u2,u3,u4}
b) Does the vector u=(2,-1,4) belong to U. Justify!
c) Is it true that U = span{ u1,u2,u3} justify the answer!

find the basis and dimension for the span of each of
the following sets of vectors.
a={[2,-1,1],[0,0,0],[-4,2,-2],[6,-3,3]}
basis=
dimension=
b={[3,3,3],[9,9,10],[21,21,23],[-33,-33,-36]}
basis=
dimension=

(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?

(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

Consider the vector: u1=(1,1,1), u2= (2,-1,1), u3=(3,0,2),
u4=(6,0,4)
a)Plot the dimension and a basis for W= span {u1,u2,u3,u4}
b)Does the vector v= (3,3,1) belong to W? justify the
answer.
c) Is it true that W= span{ u3,u4} Justify answer.

a) Find a basis for W2 = {(x, y, z) ∈ R 3 : x + y + z = 0} (over
R). Justify your answer. What is the dimension of W2?
b) Find a basis for W4 = {(x, y, z) ∈ R 3 : x = z, y = 0} (over
R). Justify your answer. What is the dimension of W4?

3. Let V and W be ﬁnite-dimensional vector spaces over ﬁeld F
with dim(V) = n and dim(W) = m, and
let φ : V → W be a linear transformation. Fill in the six blanks
to give bounds on the sizes of the
dimension of ker(φ) and the dimension of im(φ).
3. Let V and W be ﬁnite-dimensional vector spaces over ﬁeld F
with dim(V ) = n and dim(W) = m, and
let φ : V → W...

We have learned that we can consider spaces of matrices,
polynomials or functions as vector spaces. For the following
examples, use the definition of subspace to determine whether the
set in question is a subspace or not (for the given vector space),
and why.
1. The set M1 of 2×2 matrices with real entries such that all
entries of their diagonal are equal. That is, all 2 × 2 matrices of
the form: A = a b c a
2....

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