5. Let x, y be vectors in R4, and let L be the set L = {ax +By: a, ß ER, a + B = 1}. (a) Verify that L is an affine subspace of R4 (i.e., if w₁, W2 are in L then so is any affine combina- tion). (b) Suppose x = [1 2 3 4]" and y = [0 1 2 3]¹. Find a vector u and a matrix A such that L = u + null(A).

Q: 3. Let W = M2x2(C) with zero determinant. Show that W is not a subspace of V = M2x2(C).

A:

Q: 2b Let W = { "] | a, b e R}. a. Find a basis for W. (Your work here proves that W is a subspace of…

A: Given below the detailed solution

Q: 15. Let V be the set of those polynomials ax² + bx + c € P₂ such that P2 a + b + c = 0. Is V a…

A: Let V be the set of those polynomials ax2+bx+c∈P2 such that a+b+c=0. To determine whether V is a…

Q: The integer n = 12700140x3243243 is missing the digit x. a) If n congruent to  3 (mod 9), then what…

A: According to the given information, Consider the given integer n:

Q: 2. Let a, B, y be three vectors in the inner product space V½ (R) with the standard inner product…

A:

Q: 4. Let W be the subspace of R³ spanned by w the orthogonal projection of 8- onto W. - (2/3) 1/3 and…

A: orthonormal vectors:- A set of non zero vectors v1, v2, v3,......, vn is called orthonormal if…

Q: 11. Let W be the set of all vectors of the form 5 b + b 2 с where b and care arbitrary. Find vectors…

A:

Q: Let fi = 1+ 2x + 3x², f2 = 2+ 3x + 4x?. B = (f1, f2) is a basis {ao + a1x + a2x²|ao – 2a1 + a2 = 0}…

A: Given f1=1+2x+3x2 and f2=2+3x+4x2. And B=f1,f2 is a basis for the subspace V=a0+a1x+a2x2|a0−2a1+a2=0…

Q: 13 - Which one of the followings is NOT the subspace of R? ? =O W = {(a,2a): a e R} W = {(a, a + 1):…

A:

Q: Let V be a general vector space and u, v € V be linearly independent. (a) Show that the vectors x =…

A:

Q: . Let u₁ = (1, -2, -1), U₂ = (2, 2,-2), and v = = (0, 0, 1). v is not in the subspace W spanned by…

A: In this question, a vector that is orthogonal to a given vector is to be determined. A non-zero…

Q: 8. Let V={(x,y)x,y € R}. Define addition on V as follows: (x»x)+(x,y)=(x +2x, 1,y+Y,+2) Find the…

A: To find zero vector of V

Q: 9. Define the terms linear combination, linear dependence and linear independence as applied to…

A: The objective of the question is to define the terms linear combination, linear dependence, and…

Q: 1. State if the following statement is true or false. The set of vectors B={u= (1.1.4);v=(2,0.2)…

A:

Q: 8. Let 1 2 +--+- () -- () ---(3) (:). at = and = 3 4 7 b3 (a) Find conditions on b₁,b2, b3 such that…

A:

Q: Using the information in the table below, find the occupancy rate for the years 2021 to 2024 using…

A: Given α=0.3. Let At represent the actual values of the occupancy rate and Ft represent the forecast…

Q: 5. Let S {V₁, V₂, ₁), n 22 be a set of nonzero vectors of V. Each of the following statements is…

A:

Q: Is v = |1a vector in the span of S = }|2| Use Gaussian elimination to support your answer.

A: To check v=111 a vector in a span of S=123, 012, -201 or not.

Q: ow that if (V, O, •) is a vector space then: = 0 for every u in V. (Use the definition of the zero…

A:

Q: 9. If S and T are finite spanning sets for a vector space V, then S and I have the same number of…

A:

Q: C along with the following operations defined: (a1, a2) + (b1, b2) =(a1 +b1 + 1, az + b2 +1) C •…

A:

Q: 5. Let M3 - {{: ahader} M22 = {" a la,b,c, de R Show that M22 with the usual matrix addition and…

A:

Q: Let V be a finite-dimensional R-vector space. Let 3 and 3' be two ordered bases for V, and let Q =…

A:

Q: 18. Let V= {(a1, a2): a1, a2 € R}. For (a₁, a2), (b1,b2) EV and c € R, define (a1, a2) + (b₁,b2) =…

A: An operation vector addition ‘ + ‘ must satisfy the following conditions:Closure : If x and y are…

Q: 7.4Find a subset of the vectors v= (1, -2, 1, -1), v2= (0, 1, 2, –1), vz= (0, 1, 2, –1) and v,= (0,…

A:

Q: 5.2. Prove that if V is a vector space having dimension n, then a system of vectors V1, V2, ..., Vn…

A:

Q: If W, and W, are two subspaces of a vector space V (F) then prove that WinW, is also a subspace of V…

A:

Q: a) Supppose H is a 2 dimensional subspace of R³ with a basis {b₁,b2}, and y is a vector given by b₁…

A:

Q: Show that V = W₁ © W₂ for the given vector space V and the given subspaces W₁ and W₂. (a) V = R³, W₁…

A:

Q: 5. P2 denotes the set of all polynomials of degree ≤ 2, in a single variable x, with real number…

A:

Q: Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify your…

A:

Q: 33. Let V - P3 be the vector space of polynomials of degree <3, and let p₁(x) = (2-x)² and p₂(x) =…

A:

Q: 6. Vector fields V and W are defined by V = (2x – 3y + z, –3x – y + 4z, 4y + z) W = (2x – 4y – 5z,…

A: