Question

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?*

Answer #1

Determine whether the given set ?S is a subspace of the vector
space ?V.
A. ?=?2V=P2, and ?S is the subset of ?2P2
consisting of all polynomials of the form
?(?)=?2+?.p(x)=x2+c.
B. ?=?5(?)V=C5(I), and ?S is the subset of ?V
consisting of those functions satisfying the differential equation
?(5)=0.y(5)=0.
C. ?V is the vector space of all real-valued
functions defined on the interval [?,?][a,b], and ?S is the subset
of ?V consisting of those functions satisfying
?(?)=?(?).f(a)=f(b).
D. ?=?3(?)V=C3(I), and...

Prove that the singleton set {0} is a vector subspace of the space
P4(R) of all polynomials of degree at most 3 with real
coefficients.

If V is a vector space of polynomials of degree n with real
numbers as coefficients, over R, and W is generated by
the polynomials (x 3 + 2x 2 − 2x + 1, x3 + 3x 2 − x + 4, 2x 3 +
x 2 − 7x − 7),
then is W a subspace of V , and if so, determine its basis.

4. Prove the Following:
a. Prove that if V is a vector space with subspace W ⊂ V, and if
U ⊂ W is a subspace of the vector space W, then U is also a
subspace of V
b. Given span of a finite collection of vectors {v1, . . . , vn}
⊂ V as follows:
Span(v1, . . . , vn) := {a1v1 + · · · + anvn : ai are scalars in
the scalar field}...

Verify this axiom of a vector space.
Vector space:
A subspace of R2: the set of all dimension-2 vectors
[x; y] whose entries x and y are odd integers.
Axiom 1:
The sum u + v is in V.

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....

Suppose V is a vector space over F, dim V = n, let T be a linear
transformation on V.
1. If T has an irreducible characterisctic polynomial over F,
prove that {0} and V are the only T-invariant subspaces of V.
2. If the characteristic polynomial of T = g(t) h(t) for some
polynomials g(t) and h(t) of degree < n , prove that V has a
T-invariant subspace W such that 0 < dim W < n

Let P2 denote the vector space of polynomials in x with real
coefficients having degree at most 2. Let W be a subspace of P2
given by the span of {x2−x+6,−x2+2x−1,x+5}. Show that W is a proper
subspace of P2.

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...

A vector space V and a subset S are given.
Determine if S is a subspace of V by determining
which conditions of the definition of a subspace are satisfied.
(Select all that apply.)
V = C[−4, 4] and S = P.
S contains the zero vector.
S is closed under vector addition.
S is closed under scalar multiplication.
None of these

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