1. Show that the set of all polynomials of deg=2 is not a vector
space over reals.
can this be fixed, can we have a set of polynomials that is a
vector space over reals?
2. Show that the set of 2x2 matrices with m_22 = 1 is not a
vector space over reals.
3. Show that the set of infinitely-differentiable real functions
is a a vector space under pointwise function addition, and
pointwise scalar multiplication as defined in class,...
Let P2 be the vector space of all polynomials of
degree less than or equal to 2.
(i) Show that {x + 1, x2 + x, x − 1} is a basis for
P2.
(ii) Define a transformation L from P2 into
P2 by: L(f) = (xf)' . In other words,
L acts on the polynomial f(x) by first multiplying the function by
x, then differentiating. The result is another polynomial in
P2. Prove that L is a linear transformation....
Let PN denote the vector space of all polynomials of
degree N or less, with real coefficients. Let the linear
transformation: T: P3 --> P1 be the second
derivative. Is T onto? Explain. Is T one-to-one? What is the Kernel
of T? Find the standard matrix A for the linear transformation T.
Let B= {x+1 , x-1 , x2+x , x3+x2 }
be a basis for P3 ; and
F={ x+2 , x-3 } be a basis for P1 ....
Consider the vector space P2 of all polynomials of degree less
than or equal to 2 i.e. P = p(x) = ax + bx + c | a,b,c €.R
Determine whether each of the parts a) and b) defines a subspace in
P2 ? Explain your answer. a) ( 10 pts. ) p(0) + p(1) = 1 b) ( 10
pts.) p(1) = − p(−1)
S_3 is the vector space of polynomials degree <= 3. V is a
subspace of poly's s(t) so that s(0) = s(1) = 0. The inner product
for two poly. s(t) and f(t) is def.: (s,f) = ([integral from 0 to
1] s(t)f(t)dt). I would like guidance finding (1) an orthogonal
basis for V and (2) the projection for s(t) = 1 - t + 2t^2. Thank
you!
QUESTION 1
Vector Space Axioms
Let V be a set on which two operations, called vector addition
and vector scalar multiplication, have been defined. If u and v are
in V , the sum of u and v is denoted by u + v , and if k is a
scalar, the scalar multiple of u is denoted by ku . If the
following axioms satisfied for all u , v and w in V and for all
scalars k...
Let P denote the vector space of all polynomials with real
coefficients and Pn be the set of all polynomials in p
with degree <= n.
a) Show that Pn is a vector subspace of P.
b) Show that {1,x,x2,...,xn} is a basis
for Pn.
3. We let ??(?) denote the set of all polynomials of degree at
most n with real coefficients.
Let ? = {? + ??3 |?, ? ??? ???? ???????}. Prove that T is a
vector space using standard addition and scalar multiplication of
polynomials in ?3(?).
Let V be the 3-dimensional vector space of all polynomials of
order less than or equal to 2 with real coefficients.
(a) Show that the function B: V ×V →R given by B(f,g) = f(−1)g(−1)
+ f(0)g(0) + f(1)g(1) is an inner product and write out its Gram
matrix with respect to the basis (1,t,t2).
DO NOT COPY YOUR SOLUTION FROM OTHER SOLUTIONS
Explain if the set below is a vector space given standard
operations.
The set of all even functions defined on R with addition and scalar
multiplication defined as follows:
1.) (f+g)(x) = f(x) + g(x) (addition)
2.) (cf)(x) = cf(x)