In: Advanced Math

Recall P2(t) is the set of polynomials of order less than or equal to 2. Consider the the set of vectors in P2(t).

B={t^2,(t−1)^2,(t+1)^2}

(a) Show B is a basis for P2(t).

(b) If E={1,t,t^2}is the standard basis, calculate the change of basis matrices PE→B and PB→E

(c) Given v= 2t^2−5t+ 3, find its components in B

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 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

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.

prove that 2/pi is less than or equal to (sinx)/x which is less
than or equal to 1. for x is in (0,pi/2]

8. The cardinality of S is less than or equal to the cardinality
of T, i.e. |S| ≤ |T| iff there is a one to one function from S to
T. In this problem you’ll show that the ≤ relation is transitive
i.e. |S| ≤ |T| and |T| ≤ |U| implies |S| ≤ |U|.
a. Show that the composition of two one-to-one functions is
one-to-one. This will be a very simple direct proof using the
definition of one-to-one (twice). Assume...

Consider a value to be significantly low if its z score less
than or equal to −2 or consider a value to be significantly high if
its z score is greater than or equal to 2.
A data set lists weights (grams) of a type of coin. Those
weights have a mean of 5.13125 g and a standard deviation of
0.05783 g. Identify the weights that are significantly low or
significantly high.

Consider a value to be significantly low if its z score less
than or equal to minus−2 or consider a value to be significantly
high if its z score is greater than or equal to 2.
A test is used to assess readiness for college. In a recent
year, the mean test score was 20.8 an the standard deviation was
5.3. Identify the test scores that are significantly low or
significantly high.
What test scores are significantly low? Select the...

solev tan^2 x =1 where x is more than or equal to 0 but x is less than or equal to pi

Let n greater than or equal to 2 and let k1,...,kn be
positive integers. Recall that Ck1,..., Ckn denote the cyclic
groups of order k1,...,kn. Prove by induction that their direct
product Ck1×Ck2×....×Ckn is cyclic if and only if the ki's are
pairwise coprime which means gcd(ki,kj)=1 for every i not equals j
in {1,...n}.

For each of the following, fill in the blanks with "Less than",
"Greater than", or "Equal to"
*) A gas flows through a one-inlet, one-exit control volume
operating at steady state with no internal irreversibilities, Qcv =
0. Heat transfer at a rate Qcv takes place only at a location on
the boundary where the temperature is Tb. The specific entropy of
the gas at the exit is _____ than the specific entropy of the gas
at the inlet.
*)...

ADVERTISEMENT

ADVERTISEMENT

Latest Questions

- The Great Vowel Shift (GVS) is a well-studied sound change phenomenon which occurred between 1400 and...
- 1. Suppose that firm B makes rugs and that its fixed costs are 10,000 per month....
- Consider a two-dimensional universe where Maxwell's equations are valid (their corresponding two-dimensional). Derive a wave equation.
- a) What are the assumptions about the competitor's reaction to the price changes underlying the shattered...
- Write scores of 20 students on a quiz to a file. The file should be named...
- Describe the ground-breaking study of Myers and Sperry on split-brain cats. Explain why it has been...
- Do you think that the intellectual property laws on the network (worldwide network) will become less...

ADVERTISEMENT