1.F.2 Let B = {1, x, x², x³} be the standard basis of P<3. a) What are the coordinates of 5³ – 2x+1 in the standard basis of P<3? Warning: Don't forget, comes LAST in this ordered basis. b) Use coordinate vectors (in the standard basis) to test the linear independence of {1 – 2x² – r³, x + 2x³, 1+ x – 2x²}. c) Use coordinate vectors (in the standard basis) to test the linear independence of {(x–1)³, (x- 2)², (x – 3)(x² – 8x +11)}.

Q: Prove that the vectors (1. 1. 0) (1 2 3) and (2. –1 5) form a basis for R3

A:

Q: Let B= = (1, +2, (x - 1)², 42³) be an ordered basis for P3. Find the coordinate vector of f(x) = 1x³…

A:

Q: Write down a basis of R 4 where you don't use any of the standard unit vectors (e1, ... , e4). Show…

A:

Q: B = (1, x+1, x² + x) forms an ordered basis for P3. Give the B-coordinate vector of each of the…

A: The coordinate vector of v with respect to  is given by such that .(a) Given .Comparing the…

Q: Consider the vector v = 1 + 4t in terms of the standard basis for P2, {1, t, t²} and new basis B =…

A:

Q: - Linear maps Consider the following three vectors in R³: 1 0 --0) --(9)· --0 V1 = 1 V3 = = -1 1 (a)…

A:

Q: 3. Prove that the vectors u = (1, 1, 1), u = (1, 2, 1), and (1,3,-1) form a basis in R³. Find the…

A:

Q: Consider the system of vectors (1,2, 1, 1)", (0,1,3, 1)", (0,3,2,0)", (0,1,0,0)". a) Prove that it…

A:

Q: Show that the vectors a = (1,0,-1), az = (1,2,1), az = (0,-3,2) form a basis of R. Express each of…

A: It is given that

Q: Explain why S is not a basis for R². S = {(-6,4)} OS is linearly dependent. OS does not span R². O S…

A: Given: S=-6, 4. We have to explain why S is not a basis for R2.

Q: (d) Show that the polynomials P₁ = 3 + 2x + x², P₂ = 2x+3x², p3= −2+x=2x² form a basis of the vector…

A: Using 1+2x-x^2=ap1+bp2+cp3 find a b c

Q: a) Let q(t) = 4t2 + 2t + 9 be an element of P2. Write the coorfinate vector of q(t) relative to…

A: We have to find the followings ; (a) We have to find the coordinate vector of  q(t) = 4t2+2t+9 with…

Q: Show that S = {t+1,t-1,(t-1)2} be a basis of P2, the vector space of all second-degree real…

A:

Q: Consider the three basic vectors: (2,1,0,-1),(-1,1,1,1),(2,7,4,1) a. give two vectors that are in…

A: Given the three basic vectors:                           (2,1,0,-1),(-1,1,1,1),and (2,7,4,1) We have…

Q: Let S = {ax? + bx + a : a, b ɛ R}. Then a basis and the dimension of S are (a) basis: x2, x + 1;…

A:

Q: Show that the set S= {(-2,-4), (1, 2)} is the basis of R2 or not?

A:

Q: 3) Let C be a basis that contains vectors (1 + (-1) x + 1x² + 1x³), (1 + 2x + 2x² + (−1) x³), (1 +…

A:

Q: The vectors v' 1 = (4,5,3), v 2 = (8, r, 6) and v’ 3 = (6,7,8) form a basis of R for any real value…

A:

Q: Show that the vectors (1, 1, 2), (1,2,5), (5,3,4) of R'(R do not form a basis of R'(R).

A:

Q: How do I find a basis for {(2c, c, b)} | b, c ∈ R}? I tried writing it as a linear combination and I…

A:

Q: Find the vector x determined by the given coordinate vector [x]g and the given basis B. 8-{8}-(-6) B…

A:

Q: Find the coordinate vector [x]B of the vector x relative to the given basis B. 23) b1 =| b2 and B =…

A:

Q: b) Find the value of k such that (-1,1,1), (1,1,1), (1, -1, k) form a basis of R²

A:

Q: Question 4: [2 Mark] Determine whether or not the given vectors in R" form a basis for R": a) v₁ =…

A:

Q: Show that B = {v1=(1,0,0), v2=(3,7,-2), v3=(0,4,1)} is a basis for R3

A:

Q: B = (f1, f2, f3) is a basis for R2[] where fi = 1+ 2x + a², f2 = 2+ 3r + 2x², f3 = 2 + x +3x². If…

A: The correct option is option (e) 1 + 2x²   The detailed solution is as follows below:

Q: = √1 = Let 4] 2 V2 - 2 and 3 = 2 -7 Explain that B = (V1, V2, V3) forms a basis of a 3-dimensional…

A:

Q: Problem 10. Consider the following set of polynomials in P₁: S = {1+ 3t² +t³, 1+t+ 2t³ + 3t¹, t+t² +…

A: We know that two vectors are linearly dependent if one vector can be written as a scalar multiple of…

Q: ) Consider the vectors -- () --- () --- () a= = Do these vectors form a basis of R³? If yes, prove…

A: Please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: 2.) Determine whether S = is a basis for R³. If it is write u = (6,3,6) as a linear combination of…

A:

Q: If B= {(1,1,0), (0,1,1), (−1,0,2)} is a basis of R³, then the coordinate vector of v= (8, -5, 3) in…

A:

Q: If B = {(1,1,0), (0,1,1), (-1,0,2)} is a basis of R³, then the coordinate vector of v = (2,0,1) in…

A: Solve the following

Q: 2. Consider the following sets of vectors. Show that each set (i) contains the zero vector, (ii) is…

A: The given sets are (a) W is the set of vectors of the form 3s, -2s, s. We have to show that  i Each…