5. Let V = {ax¹ + (b − c)x² + 3ax +c+2b: a, b, c ≤ R} ≤ P4. Construct an isomorphism from V to R³. Hint: Use Theorem 1.6.7 (p. 27) and the box in sec. 1.3.3 (p. 15). Make sure to justify if you state that something is a basis.

Q: Recall that P, is the collection of all polynomials of degree 2 or lower. Let S be the collection of…

A: Given that S=-3t2+t, t2-3t-4 (a) The linear combination of these vectors is,…

Q: The subset { (1,-2), (2,9), (-4,3) } of R is O a. linearly independent O b. linearly dependent O c.…

A: Given; {(1,-2),(2,9),(-4,3)}

Q: Let M1 M2 M3 = and 3 4 A1 = A2 A3 : Let B {M1, M2, M3} and W = span(B). It can be shown that B' {A1,…

A: The solution is given as follows

Q: 8. Consider the basis * B = {1 + x + x², x + x², x?}. for P2. Let p(x) = 2 + x+ 2r². Then [p(x)]8 =…

A:

Q: Let 0 M1 M2 M3 and 6. 3 5 A1 A2 A3 Let B {M1, M2, M3} and W = span(B). It can be shown that B' {A1,…

A:

Q: 20. Let B be a basis for R". Prove that the vectors V₁, V2, . ...9 span R” if and only if the…

A: To prove the statements, will start by defining the notation used:- Let be a basis for , where each…

Q: 1) Find the value of c such that the set is a basis foI

A: Given, the set                   20-1,310,  1-1   0   is  a basis for R3

Q: { 1 + x + x², x + x², x²}. %3D for P2. Let p(x) = -1 +x + 2x2. Then [p(x)]B = %3D %3D

A: The given basis is B=1+x+x2,x+x2,x2 for p2. let p(x)=-1+x+2x2 We have to find,  p(x)B=?

Q: Let 4. -2 9. where c ER. What is the only value of c for which the set basis for R?

A:

Q: Is the set S=I21",[ - 1 31], [2 4 5 ]] [-1 -1 3 1 a basis tor R".

A:

Q: 2. Consider the following mappings: • A€ L(R³) such that A(@1, a2, a3) = (a1 + a2 + a3, R CIP3) cuch…

A: (a). 1. Aa1,a2,a3=a1+a2+a3,0,0 Consider b1,b2,b3, Ab1,b2,b3=b1+b2+b3,0,0 1.…

Q: Exercise 5.2.1: Here are two relations: R(A, B): {(0,1), (2,3), (0,1), (2,4), (3,4)} S(B,C): {(0,1),…

A: Given: a. Compute πA+B, A2, B2R A B A+B A2 B2 0 1 1 0 1 2 3 5 4 9 0 1 1 0 1 2 4 6 4 16…

Q: 4. Consider the set X = 1 which of the following is a basis for span(X) Select one: a. 3. b.…

A: We will find out the required basis for given span(X).

Q: 0 1 0 S = {(1 3)·(¦¯¯^¹)· (² ¦ ). ( -º2³)} 03 4 0 23 2 is independent or not and if it is not,…

A: Recall the definition of linear independence i.e non-zero span of vectors must be non-zero. If a…

Q: 8. Find the max and min of |(z + 1)²e¯²| in the disk |z| ≤ 1.

A:

Q: 3. Determine which of the following sets are bases for P2 (R). (a) {-1-x+2x², 2+x-2x², 1-2x + 4x²}…

A:

Q: 2. Determine if the set is a basis of R3 [t] . {t² + 2t -3, t² + 5t, 2t² — 4}

A:

Q: 16. Determine whether the given set forms a basis for P3. {-1 -r', 2r + 2r", –2r + r – a*, 1+ r² +…

A: Follow the steps.

Q: Show that in an n-dimensional vector space X, the representation of any x as a linear combination of…

A: The given problem is related with linear algebra. We have to show that in an n-dimensional vector…

Q: 3. Prove whether { 221 2} is or is not a basis for R³. Be sure to discuss the requirements in the…

A:

Q: 3. Define a mapping T: P3-P3 by. T(f(x))=f(x) Please find the nullity of T. (Hint: Still dx by using…

A:

Q: 0,1,0) R3 Ba

A:

Q: 5. Consider the basis aj = (1, 2, 3), az = (2, 4, 5), and az = (3, 5, 6) for R] × 3. Consider t E…

A: Given α1 =1, 2, 3, α2 =2, 4, 5, α3 =3, 5, 6 is the basis of R1×3 So let α = α1, α2, α3 Let τ = α…

Q: Show that the following set is not a basis of R4: 1 3 2 2 3 15 1 -3 2 -17 2 -3 13

A:

Q: 53. Consider the basis B of R² consisting of the vectors [3] [F]g = | 1 and We are told that i 93…

A:

Q: Suppose ū (5, 3, 0) and v = (3, 1, –1). The projection of ū along ở is

A:

Q: 9. Find a basis for the hyperplane a where a-(2, 1, -3).

A: Any two linearly independent vectors which are orthogonal to the given a are forms a orthogonal…

Q: 10. Determine which sets are a basis for R². (a) (b) © {[2] [9] [17]} {8.8} (Ⓒ) {B} {[]+[2]}

A: To identify which of the given set is basis for R2

Q: DETER MINE WHETH ER TiHE SET Å IS A BASIS FOr IR9 2 %3D -2 EXPLAIN YOUR AN SWER MN- -m.

A: The given set S is not a basis for R^4.

Q: Show how you find the Kernel and Image of T. Then verify dim Ker T and dim Im I you obtained by…

A: In this question, we have to find the Kernel and Image of any given linear transformation T and…

Q: Let S = {v1, V2, V3, V4, V5} be the subset of M14 with 2 -1 1 1: ] V1 V2 4 2 V3 2 3 4 -1 1 -1 |, ]…

A: We have to find a basis for span S as a subset of S.

Q: Prove (A → A) → A in H. Note: In addition to the axioms and rule of inference of H, you may use any…

A: ↦(⇁A→A)→A 1. Assume (⇁A→A)→A (Assumption) 2. Assume ⇁A→A(Assumption)

Q: (a) Suppose that V =Span{ 2 2 8 Determine whether each of the following sets is a basis for V: 2 | 2…

A: