. If {x1, X2, X3} is a basis of V3 (R), show that {x, +x2, X2 + x3, X3 + X1} _is also a basis of V3 (R).

Q: Let 9. A = 3 9 4 6. -3 2 -2 Find an orthonormal basis of the kernel of A.

A:

Q: Explain why S is not a basis for R³. S = {(3, 1, 0), (4, 2, 1), (2, 0, -1)}

A: Linear dependent: - If a non-trivial combination of these vectors equals the zero vector, the…

Q: Determine whether S is a basis for P3. S = {-3 + t³, 2t2, 2 + t, 1 + 2t + 2t2 + t3} O is a basis of…

A: A set S is a basis of a vector space V iff dimension of V= number of linearly independent vectors of…

Q: +t, 2t +t, 6t} O sis a basis of P O sis not a basis of P3.

A:

Q: 4. Determine if B = {1,2 – x,3 - x2, r + 2x2} is a basis for P.

A: Given

Q: Do the polynomials x2 + 1, x? – 1, 2x – 1 form a basis in P2

A:

Q: Determine whether S is a basis for P3. S= {5 – 312 + t2, -4 + t2, 3t + t³, 6t} O is a basis of P3. S…

A:

Q: Determine if S is a basis of P3. S = {1-t, 2t + 3t2, t2 - 2t3, 2+t3}

A:

Q: The coordinates of p(x)=5+3x+x² relative to the basis {1+x,2-x2,3+x+x²} is O (-1,1,2) none (0,1,1) O…

A:

Q: Let S = {x; 2 – x; x² – 2x}. A. S is a basis of P2 B. S is a basis of P C. None of the above

A: Let S=x, 2-x, x2-2xSince, we know that If V is a vector space of dimension n  (say) and…

Q: Find the basis of the null space of A. 3 1 1 2 A = 4 1 2 1 1 –1 1 1 1 -1 [2 2 1 2 1 2

A: Step 1: Transform the matrix to the reduced row echelon form

Q: The set B = {2 – 4x?, 8+ 3x – 16x2, 21 + 9x – 40x²} is a basis for P2. Find the coordinates of p(x)…

A:

Q: 1

A:

Q: 2. Find the basis for the nullspace of A, and the nullity and rank of A

A: This is a problem of linear transformation.

Q: ii) Show that S = {(t – 1)³, (t – 1)², (t – 1), 1} is a basis of P3(t) where P3(t) stands for the…

A:

Q: The set B = {4 – 2x2, 20 + x – 10x2, 28x2 – (55 + 3x)} is a basis for P2. Find the coordinates of…

A:

Q: which is the collection of all polynomials of degree ≤ 3. Write out the standard basis for P2? What…

A:

Q: Determine whether S is a basis for P3. S = {3 - 4t2 + t3, -3 + t?, 2t + t³, 6t} O is a basis of P3-…

A:

Q: Which of the following sets of polynomials form a basis for P2?

A:

Q: Let B={1, x, x², x³} be a basis for P3, and let P={P₁, P2, P3, P4} be the set of polynomials given…

A:

Q: S = (1 + x, 1 – 2x) is not a basis for P,(the space of polynomials of degree less than 2) Select…

A: False.

Q: The set B = {2 + x², 6 + x + 3x?, 19 + 2x + 9x²} is a basis for P2. Find the coordinates of p(x) (39…

A:

Q: Let B = {x²+x,-2x² + 3x + 3, 2x² – x – 2} be a basis of P2(R)

A: Given,            B=x2+x,  -2x2+3x+3,  2x2-x-2

Q: Let V = P4(R). Find a basis of V containing the polynomials p1(x) = 1 + 2x-x 2+3x 3+2x 4 , p2(x) =…

A: The detailed solution of the problem is given in the next steps.

Q: Which of the following is NOT a basis of  P2       {1 + X2 ,  X,  1}       {2,  2 + X2  , X2   }…

A: Given the sets of vectors we need to check which one is not a basis of P2.

Q: Determine whether S is a basis for P3. S = {5 – 3t2 + t3, -2 + t2, 3t + t³, 5t} O is a basis of P3.…

A:

Q: 2: Consider B= a) Show that B is a basis for R³.

A:

Q: 4 2. If a basis for R3 is B = -3 and given [x]B find i in the standard basis.

A:

Q: Suppose v1, v2, v3, V4 is a basis of V. Prove that vi + v2, V2 + V3, V3 + V4, V4 is also a basis of…

A: we are given that v1,v2,v3,v4 is basis of V. we have to prove that v1+v2,v2+v3,v3+v4 ,v4  is also…

Q: Determine whether S is a basis for P3. S = {3 – 2t2 + t3, -2 + t², 3t + t³, 5t} S is a basis of P3.…

A:

Q: The set B = {1– 2t + t2,3 – 5t + 4t² , 2t + 3t²} is a basis of P2. Suppose C = {c1, C2, C3} is…

A:

Q: is linearly independent. spans R3 5 is a basis for R³. None of the above.

A: Given set of vectors are : S=003.1,-2.74.90,-2.74.93.1 in R3 Now need to select which statements are…

Q: Find the basis of the null space and basis of the column space.

A: Consider the provided matrix A, First reduce the matrix A into the row echelon form,

Q: Show that the polynomials P1 = 1+x + x² P2 = x + x² P3 = x2 form a basis for P2.

A:

Q: Let B = {1+x, 1– 3x?, x + 6x²} and p(x) = 12 + 8x + 9x?. You are given that B is a basis for P2. If…

A:

Q: Let B={1, x, x², x³} be a basis for P3, and let P={P1, P2, P3, P4} be the set of polynomials given…

A:

Q: Consider B find [p(x)]B. {x2 + x, 2 + 2x, 1} an ordered basis of P2(R). If p(x) = 4.x² 2x – 6, %3D

A:

Q: Determine whether S is a basis for P,. S = {-3 + t, 2t?, 3 + t, 3 + 2t + 2t2 + t°} O is a basis of…

A:

Q: Determine a basis B for P2(R) whose elements all have the same degree. Prove that B is a basis.

A:

Q: 60. In R4, given a nullspace V = {(x1; X2; X3; X4) E R4|¤1 – *2+ x3 = 0 A x2 + x3 + x4 = 0}. %3D…

A: see below the answer and explanation

Q: Let B = {x – 2, x² – 2,4x}. Which of the following statements is true? B is a basis of P3 B is a…

A:

Q: Determine whether it is a basis. S= {4 – x, x³, 6x², x³ + 3x, 4x – 1}

A: No it is not a basis  see below the explanation

Q: Let P2 be the polynomials of degree no more than 2. Is {x2+x+1,2x+1,3x2+1} a basis for P2 ?

A:

Q: Explain why S is not a basis for P2.S = {1 + x2, 1 − x2}

A: Given: S = {1 + x2, 1 − x2}

Q: The set B = {x² – 2, 4x² − (8+3x), 15x² − (28 + 9x)} is a basis for P₂. Find the coordinates of p(x)…

A: Result: Let B be the basis of P2 and let p(x)∈P2. Then the coordinates of p(x) relative to the basis…

Q: Determine whether or not S = {x² +1, x+ 2, –x² +x} is a basis for P2.

A:

Q: Let Bo = {1, X, X²} be the canonical base of R2[X] and put %3D 1+X+2X = 1+X+X2 = 1+2X +X² E1 !! E2…

A: Note:- As per our guidelines, We’ll answer the first question since the exact one wasn’t specified.…

Q: Determine whether S is a basis for P S = {5 - 4t2 + t, -4 + t2, 2t + t, 5t} O is a basis of P2. O s…

A: