Question

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.

Answer #1

if u have any questions please comment

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;

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....

Determine if the given set V is a subspace of the vector space
W, where
a) V={polynomials of degree at most n with p(0)=0} and W=
{polynomials of degree at most n}
b) V={all diagonal n x n matrices with real entries} and W=all n
x n matrices with real entries
*Can you please show each step and little bit of an explanation
on how you got the answer, struggling to learn this concept?*

Let Mn be the vector space of all n × n matrices with real
entries. Let W = {A ∈ M3 : trace(A) = 0}, U = {B ∈ Mn : B = B t }
Verify U, W are subspaces . Find a basis for W and U and compute
dim(W) and dim(U).

Answer all of the questions true or false:
1.
a) If one row in an echelon form for an augmented matrix is [0 0 5
0 0]
b) A vector b is a linear combination of the columns of a matrix A
if and only if the
equation Ax=b has at least one solution.
c) The solution set of b is the set of all vectors of the form u =
+ p + vh
where vh is any solution...

Let V be a vector space of dimension n > 0, show that
(a) Any set of n linearly independent vectors in V forms a
basis.
(b) Any set of n vectors that span V forms a basis.

(1 point) Which of the following subsets of {R}^{3x3} are
subspaces of {R}^{3x3}?
A. The 3x3 matrices with determinant 0
B. The 3x3 matrices with all zeros in the first row
C. The symmetric 3x3 matrices
D. The 3x3 matrices whose entries are all integers
E. The invertible 3x3 matrices
F. The diagonal 3x3 matrices

If A, B and C= A.B are square n×n matrices and the Nullity(C)=
0, then the columns of each of these 3 matrices span the same
vector space
.
Please give a simple answe thanks

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.

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.

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