Question

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. The set M2 of 3×3 matrices with real entries such that the entry A(1, 3) = 1.

3. The set Q of quadratic polynomials p(x) = ax2+bx+c such that p 0 (2) = 0.

4. The set C of cubic polynomials p(x) = ax3+bx2+cx+d such that p(3) = 7 and p(2) = 1. Remember: You can first check whether the vector 0 is in the set for a quick check. If it is, you still need to check whether sums and product by a scalar takes you out of the set (or not).

Answer #1

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?*

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 .

Consider P3 = {a + bx + cx2 +
dx3 |a,b,c,d ∈ R}, the set of polynomials of degree at
most 3. Let p(x) be an arbitrary element in P3.
(a) Show P3 is a vector space.
(b) Find a basis and the dimension of P3.
(c) Why is the set of polynomials of degree exactly 3 not a
vector space?
(d) Find a basis for the set of polynomials satisfying p′′(x) =
0, a subspace of P3.
(e) Find...

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.

Linear Algebra-- Subspaces of Vector Spaces
Determine whether the set W is a subspace of R^3 with the
standard operations. Justify your answer.
(a): W={(0,x2,x3): x2 and x3 are real numbers}
(b): W={(a, a-3b, b): a and b are real numbers}

Question 4. Consider the following subsets of the vector space
P3 of polynomials of degree 3 or less: S = {p(x) : p(1) = 0} and T
= {q(x) : q(0) = 1} Determine if these subsets are vectors spaces
with the standard operations for polynomials

The trace of a square n×nn×n matrix A=(aij)A=(aij) is the sum
a11+a22+⋯+anna11+a22+⋯+ann of the entries on its main
diagonal.
Let VV be the vector space of all 2×22×2 matrices with real
entries. Let HH be the set of all 2×22×2 matrices with real entries
that have trace 11. Is HH a subspace of the vector space
VV?
Does HH contain the zero vector of VV?
choose H contains the zero vector of V H does not contain the zero
vector...

5.
Let S be the set of all polynomials p(x) of degree ≤ 4 such
that
p(-1)=0.
(a) Prove that S is a subspace of the vector space of all
polynomials.
(b) Find a basis for S.
(c) What is the dimension of S?
6.
Let ? ⊆ R! be the span of ?1 = (2,1,0,-1), ?2
=(1,2,-6,1),
?3 = (1,0,2,-1) and ? ⊆ R! be the span of ?1 =(1,1,-2,0), ?2
=(3,1,2,-2). Prove that V=W.

In this question we denote by P2(R) the set of functions {ax2 +
bx + c : a, b, c ∈ R}, which is a vector space under the usual
addition and scalar multiplication of functions. Let p1, p2, p3 ∈
P2(R) be given by p1(x) = 1, p2(x) = x + 2x 2 , and p3(x) = αx + 4x
2 . a) Find the condition on α ∈ R that ensures that {p1,
p2, p3} is a basis...

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...

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