Question

Let P4 denote the space of polynomials of degree less than 4 with real coefficients. Show that the standard operations of addition of polynomials, and multiplication of polynomials by a scalar, give P4 the structure of a vector space (over the real numbers R). Your answer should include verification of each of the eight vector space axioms (you may assume the two closure axioms hold for this problem).

Answer #1

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 P(R) denote the family of all polynomials (in a single
variable x) with real coefficients. We have seen that with the
operations of pointwise addition and multiplication by scalars,
P(R) is a vector space over R. Consider the 2 linear maps D, I :
P(R) to P(R), where D is differentiation and I is
anti-differentiation. In detail, for a polynomial p =
a0+a1x1+...+anxn,
we have D(p) =
a1+2a2x+....+nanxn-1
and I(p) =
a0x+(a1/2)x2+...+(an/(n+1))xn+1.
a. Show that D composed with I...

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.

Let R[x] be the set of all polynomials (in the variable x) with
real coefficients. Show that this is a ring under ordinary addition
and multiplication of polynomials.
What are the units of R[x] ?
I need a legible, detailed explaination

Let P be the vector space of all polynomials in x with real
coefficients. Does P have a basis? Prove your answer.

Let V be the set of all triples (r,s,t) of real numbers with the
standard vector addition, and with scalar multiplication in V
deﬁned by k(r,s,t) = (kr,ks,t). Show that V is not a vector space,
by considering an axiom that involves scalar multiplication. If
your argument involves showing that a certain axiom does not hold,
support your argument by giving an example that involves speciﬁc
numbers. Your answer must be well-written.

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

8. List all irreducible polynomials with binary coefficients of
degree 4 or less. (Hint: produce a times table that shows the
minimum number of products needed.) Show these as binary numbers
(omitting the indeterminant) and as decimal numbers (interpreting
the binary number into decimal). Is 23 a prime polynomial in this
field?
9. Interpreting these decimal numbers into coefficients of
polynomials with binary coefficients, what is the product of 11 and
10 modulo 31 in GF(2^4) over P = 31?...

Let V be the set of all ordered pairs of real numbers. Consider
the following addition and scalar multiplication operations V. Let
u = (u1, u2) and v = (v1, v2).
• u ⊕ v = (u1 + v1 + 1, u2 + v2 + )
• ku = (ku1 + k − 1, ku2 + k − 1)
Show that V is not a vector space.

Let V be the set of all ordered pairs of real numbers. Consider
the following addition and scalar multiplication operations V. Let
u = (u1, u2) and v = (v1, v2).
• u ⊕ v = (u1 + v1 + 1, u2 + v2 + )
• ku = (ku1 + k − 1, ku2 + k − 1)
1)Show that the zero vector is 0 = (−1, −1).
2)Find the additive inverse −u for u = (u1, u2). Note:...

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