Question

Prove that C is a real vector space with the usual sum and scalar multiplication.

Answer #1

Consider the vector space M2x2 with the usual addition and
scalar multiplication. Let it be the subspace of M2x2 defined as
follows:
H= { | a b |
| c d | with b= c}
consider matrices A= | 1 2 | B= |-2 2 |
C= | 1 8 |
| 1 3 | , | 1 -3 | , | 4 6 |
Do they form a basis for H? justify the answer

Let S={(0,1),(1,1),(3,-2)} ⊂ R², where R² is a real vector space
with the usual vector addition and scalar multiplication.
(i) Show that S is a spanning set for R²
(ii)Determine whether or not S is a linearly independent set

Show that Mm,n with standard addition and scalar multiplication
is a vector space.

Determine whether the set with the definition of addition of
vectors and scalar multiplication is a vector space. If it is,
demonstrate algebraically that it satisfies the 8 vector axioms. If
it's not, identify and show algebraically every axioms which is
violated. Assume the usual addition and scalar multiplication if
it's not defined. V = R, x + y = max( x , y ), cx=(c)(x) (usual
multiplication.

Determine whether the set with the definition of addition of
vectors and scalar multiplication is a vector space. If it is,
demonstrate algebraically that it satisfies the 8 vector axioms. If
it's not, identify and show algebraically every axioms which is
violated. Assume the usual addition and scalar multiplication if
it's not defined. V = R^2 , < X1 , X2 > + < Y1 , Y2 > =
< X1 + X2 , Y1 +Y2> c< X1 , X2...

use the subspace theorem ( i) is it a non-empty space? ii) is it
closed under vector addition? iii)is it closed under scalar
multiplication?) to decide whether the following is a real vector
space with its usual operations:
the set of all real polonomials of degree exactly n.

Determine if W is a subspace of R^3 under the usual addition and
scalar multiplication. Either show algebraically that it is or show
how it isn't algebraically. W= {(x1, x2, x3) ∈ R^3 x1 = x2 and x2 =
2x3 }

3. Closure Properties
(a) Using that vector spaces are closed under scalar
multiplication, explain why if any nonzero vector from R2 or R3 is
in a vector space V, then an entire line’s worth of vectors are in
V.
(b) Why isn’t closure under vector addition enough to make the same
statement?
4. Subspaces and Spans: The span of a set of vectors from Rn is
always a subspace of Rn. This is relevant to the problems below
because the...

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!

Prove that the set V of all polynomials of degree ≤ n including
the zero polynomial is vector space over the field R under usual
polynomial addition and scalar multiplication. Further, find the
basis for the space of polynomial p(x) of degree ≤ 3. Find a basis
for the subspace with p(1) = 0.

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