Question

In: Advanced Math

Verify all axioms that show that the set of second degree polynomials is a vector space....

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}

Solutions

Expert Solution

First we prove that is a vector space. Then find its rank.


Related Solutions

1. Show that the set of all polynomials of deg=2 is not a vector space over...
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...
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...
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...
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_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...
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...
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...
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...
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...
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)
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT