5. Determine whether the following polynomials form a basis for P2 2:2 +1, -2, +1 -1....
(a). Determine whether the set is linearly dependent or independent. Further, if it is linearly dependent, express one of the polynomials as a linear combination of others. (b). Determine whether the set can be considered as a basis of the vector space P2, which is the set of all polynomials of degree not more than 2 under addition and scalar multiplication. (1). B = {1 – 2,1 – 22, x – x2} (Hint: Similar to the matrix case in last...
Determine whether S is a basis for R. S = {(2, 4, 3), (0,4,3), (0, 0,3)} OS is a basis for R3 S is not a basis for R3. If S is a basis for R3, then write u = (6, 8, 15) as a linear combination of the vectors in S. (Use S1, S2, and sz, respectively, as the vectors in S. If not possible, enter IMPOSSIBLE.) us
Question 4: 4. Show that the following polynomials form a basis for P3 1 - x, 1-x2 1 +x _X 5. Show that the following matrices form a basis for M22 -8 1 0 3 12 -6 -4 2 _ 13. Find the coordinate vector of v relative to the basis S = {v1, V2, V3} for R3 (a) v (2, -1 3); vi = (1,0, 0), v2 = (2, 2, 0) Vз — (3, 3, 3) (b) v (5,...
4. Determine whether the polynomials Pi = 1 + x, P2 = 1 + x2, P3 = x + 2 are linearly independent or linearly dipendent in P3.
3. [20 marks] A linear transformation T: P2 + R’ is defined by [ 2a – b 1 T(a + bt + ct?) = a +b – 3c LC-a ] (1). [6 marks] Determine the kernel Ker T of the transformation T and express it in the form of a span of basis. Further, state the dimension of Ker T (2). [6 marks) Find the range Range T of the transformation T and express the range in the form of...
1. (15 points) Prove whether the following sets are linearly dependent or independent, and determine whether they form a basis of the vector space to which they belong. s 10110 -1 ) / -1 2) / 2 1 17 ) } in M2x2(R). "11-21 )'(1 1)'( 10 )'(2 –2 )S (b) {23 – X, 2x2 +4, -2x3 + 3x2 + 2x +6} in P3(R) (the set of polynomials of degree less than 3. (c) {æ4—23+5x2–8x+6, – x4+x2–5x2 +5x-3, x4+3x2 –...
Let B be the standard basis of the space P2 of polynomials. Use coordinate vectors to test whether the following set of polynomials span P2. Justify your conclusion. 1-3t+ 2t?, - 4 + 9t-22, -1 + 412, + 3t - 6t2 Does the set of polynomials span P2? O A. Yes, since the matrix whose columns are the B-coordinate vectors of each polynomial has a pivot position in each row, the set of coordinate vectors spans R3. By isomorphism between...
Please provide answer in neat handwriting. Thank you Let P2 be the vector space of all polynomials with degree at most 2, and B be the basis {1,T,T*). T(p(x))-p(kr); thus, Consider the linear operator T : P) → given by where k 0 is a parameter (a) Find the matrix Tg,b representing T in the basis B (b) Verify whether T is one-to-one and whether or not it is onto. (c) Find the eigenvalues and the corresponding eigenspaces of the...
Let S={2,3+x,1−x2}, p(x)=2−x−x2 and V=P2 (a) If possible, express p(x)as a linear combination of vectors in S. (b) By justifying your answer, determine whether the set S is linearly independent or linearly dependent. (c) By justifying your answer, determine whether the set S is a basis for P2 Please solve it in very detail, and make sure it is correct.
Standard A1 version 5 A1. Consider the following maps of polynomials S : P2 → pl and T: P2 → pi defined b S(f)-3f(0)r + f"(O) and T)3f(0)a + (f"(o)2 Show that one of these maps is a linear transformation, and that the other map is not.