Question

Linear Algebra

Does the set of all polynomials with a_n=1 form a linear space? Explain?

Answer #1

Show that the set of sequences that satisfy the linear
recurrence equation a_n+3 − c*(a_n+2) − b*(a_n+1) − a*(a_n) = 0 is
a linear subspace of the vector space of infinite sequences. Sorry
for the clunky notation - the underscores stand to signify a
subscript.

Let H be the set of all polynomials of the form p(t) = at2 where
a ∈ R with a ≥ 0. Determine if H is a subspace of P2. Justify your
answers.

Let V be the set of polynomials of the form ax + (a^2)(x^2), for
all real numbers a. Is V a subspace of P?

Linear Algebra: Show that the set of all 2 x 2 diagonal matrices
is a subspace of M 2x2.
I know that a diagonal matrix is a square of n x n matrix whose
nondiagonal entries are zero, such as the n x n identity
matrix.
But could you explain every step of how to prove that this
diagonal matrix is a subspace of M 2x2.
Thanks.

Linear Algebra Conceptual Questions
• If a subset of a vector space is NOT a subspace, what are the
four things that could go wrong? How could you check to see which
of these four properties aren’t true for the subset?
• Is it possible for two distinct eigenvectors to correspond to
the same eigenvalue?
• Is it possible for two distinct eigenvalues to correspond to
the same eigenvector?
• What is the minimum number of vectors required take to...

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.

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 Prove that Q, the set of rational numbers, with
the usually addition and multiplication, is a field.

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

(Linear Algebra) Consider the difference equation.
yk+2 - 4yk+1 + 4yk = 0, for all
k
(a) After using auxiliary equation, the solutions have the form
rk and k(rk). Find the root, r, and show that
yk = k(rk) is a solution.
(b) Show that rk and k(rk) are linearly
independent and form the general solution of the difference
equation.

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