Question

Give an example of 4 vector spaces of dimension 3.

Answer #1

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;

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...

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.

Construct a linear transformation T : V → W, where V and W are
vector spaces over F such that the dimension of the kernel space of
T is 666. Is such a transformation unique?
Give reasons for your answer.

2. Let A be of dimension 3 × 4, B be of dimension 3 × 3, C be of
dimension 2 × 3, and D be of dimension 3 × 2. Determine which of
the following calculations are possible. If they are, give the
dimensions. If not, explain why not.
(a) (BA)C
b) CD + BA

4. Which of the follows are vector spaces? Prove or
disprove.
(a) The set {x = αz | α ∈ R, z = (4, 6)T
}.
(b) The set of all 3 × 3 matrices which have all negative
elements

What is a vector? Give an example of a vector-borne disease and
name the vector responsible for transmission.

Give an example of a ternary cyclic code of length 8
and dimension 5.
Explain why the example is valid.

1. Let T be a linear transformation from vector spaces
V to W.
a. Suppose that U is a subspace of V,
and let T(U) be the set of all vectors w in W
such that T(v) = w for some v in V. Show that
T(U) is a subspace of W.
b. Suppose that dimension of U is n. Show that
the dimension of T(U) is less than or equal to
n.

Please give a example that utilise the idea of “Vector”

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