In: Advanced Math

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!

Define a subspace of a vector space V . Take the set of vectors
in Rn such that th
coordinates add up to 0. I that a subspace. What about the set
whose coordinates add
up to 1. Explain your answers.

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

Let
V be the space of polynomials with real coefficients of degree at
most n, and let D be the differentiation operator. Find all
eigenvectors of D on V.

Verify all axioms that show that the set of second degree
polynomials is a vector space. What is the Rank?
P2 = {p(x)P | p(x) = ax^2 + bx + c where a,b,c E
R}

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....

Suppose U is a subspace of a finite dimensional vector space V.
Prove there exists an operator T on V such that the null space of T
is U.

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 ....

V is a subspace of inner-product space R3, generated
by vector
u =[1 1 2]T and v
=[ 2 2 3]T.
T is transpose
(1) Find its orthogonal complement space V┴ ;
(2) Find the dimension of space W = V+ V┴;
(3) Find the angle q between u and
v; also the angle b between
u and normalized x with respect
to its 2-norm.
(4) Considering v’ =
av, a is a scaler, show the
angle q’ between u...

(3) Let V be a vector space over a field F. Suppose that a ? F,
v ? V and av = 0. Prove that a = 0 or v = 0.
(4) Prove that for any field F, F is a vector space over F.
(5) Prove that the set V = {a0 + a1x + a2x 2 + a3x 3 | a0, a1,
a2, a3 ? R} of polynomials of degree ? 3 is a vector space over...

4. Whether P3 or the space of the polynomials of degree less
than or equal to 3 and consider T: P3 → P3, given by the derivation
T(f) = f' . For example, T (−3x 2 + 5x - 10) = −6x + 5.
(a) Prove that T is a linear transformation.
(b) Determine ker (T).
(c) Is the T transformation injective? Justify that.
(d) The polynomial g (x) = 3x^2 + 1 belongs to the image?
Justify that.

ADVERTISEMENT

ADVERTISEMENT

Latest Questions

- Model of tree growth, support with graph. (a) Discuss the biological decision rule to harvest the...
- Case for Analysis: Change at Defence Research and Development—DRDC Toronto DRDC Toronto is a research centre...
- You are a federal prosecutor in Washington, DC, assigned to the following case: Robert is a...
- Suppose the followings represent the goods market: C = C0 + c(Y-T) (Consumption Function) I =...
- To what degree is the ‘invisible hand’ working today as Smith envisioned; are today’s business activities...
- Per the text book, OB experts have proposed 12 mechanisms or levers for changing organizational culture....
- Write a function called printChList that takes as its parameters a character array, its size, and...

ADVERTISEMENT