Question

Consider W as the set of all skew-symmetric matrices of size 3×3. Is it a vector space? If yes, then ﬁnd its dimension and a basis.

Answer #1

A matrix A is symmetric if AT = A and skew-symmetric
if AT = -A. Let Wsym be the set of all symmetric
matrices and let Wskew be the set of all skew-symmetric
matrices
(a) Prove that Wsym is a subspace of Fn×n . Give a
basis for Wsym and determine its dimension.
(b) Prove that Wskew is a subspace of Fn×n . Give a
basis for Wskew and determine its dimension.
(c) Prove that F n×n = Wsym ⊕Wskew....

We can identify the set V of all 3×3-matrices (real
coefficients) with vector space R9. Show that the set of all 3 × 3
symmetric matrices is a vector subspace of V .

Let the set W be: all polynomials in P3 satisfying
that p(-t)=p(t),
Question: Is W a vector space or not?
If yes, find a basis and dimension

Is W = {y(x) = (ax+b)e^−x}where a,b ∈ R are arbitrary constants,
is a vector space? If yes, ﬁnd its dimension. Give explanations

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.

What is the highest possible dimension of a subspace of M_n
(R) (set of n×n matrices with real coefficients with its usual
vector space structure) that only contains invertible matrices (and
0) ?

Show that the set GLm,n(R) of all mxn matrices with
the usual matrix addition and scalar multiplication is a finite
dimensional vector space
with dim GLm,n(R) = mn.
Show that if V and W be finite dimensional vector spaces with
dim V = m and dim W = n, B a basis for V and C a basis for W
then
hom(V,W)-----MatB--->C(-)-------->
GLm,n(R) is a bijective linear transformation. Hence or
otherwise, obtain dim hom(V,W).
Thank you!

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).

Show that the set Vof all 3 x 3 matrices with distinct entries
also combination of positive and negative numbers is a vector space
if vector addition is defined to be matrix addition and vector
scalar multiplication is defined to be matrix scalar
multiplication

Let V be an n-dimensional vector space and W a vector
space that is isomorphic to V. Prove that W is also
n-dimensional. Give a clear, detailed, step-by-step
argument using the definitions of "dimension" and "isomorphic"
the Definiton of isomorphic: Let V be an
n-dimensional vector space and W a vector space that is
isomorphic to V. Prove that W is also n-dimensional. Give
a clear, detailed, step-by-step argument using the definitions of
"dimension" and "isomorphic"
The Definition of dimenion: the...

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