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

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

Related Solutions

Let P2 be the vector space of all polynomials of degree less than or equal to...
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...
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...
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...
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....
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...
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...
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...
solve tan^2 x =1 where x is more than or equal to 0 but x is less than or equal to pi
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...
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...
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. *)...