Let V = M2×2(F), be the vector space of all 2 × 2 matrices over the real number field. Define W1 = {(a b             c a) ∈ V: a, b, c ∈ F}, W2 = {(0 a           −a b) ∈ V: a, b ∈ F} Prove that W1 and W2 are subspaces of V, and find the dimensions of W1,W2,W1 + W2and W1 ∩ W2.

Q: Let V be the vector space of real 2 × 2 matrices with inner product (A|B) = tr(BtA). Let U be the…

A: Now , V=abcd:a,b,c,d ∈IRbe the Vector spaceof 2×2 real matricesendowed with the inner…

Q: Let 01, 02, 03 be three nonzero vectors in a vector space V. Suppose that V1, V2, V3 are orthogonal…

A:

Q: 20 5 -1 6 2 The null space for the matrix 4 4 -1 5 4 1 0 11 is spanned by the vector 10

A: Definition of Null Space of a Matrix The null space of an m × n matrix A is the set of all the…

Q: In a vector space V, prove that (-1)v= -v for all v E V.

A:

Q: Let V be the vector space of all (3×3) matrices and W be the set of all (3×3) matrices A for which…

A:

Q: A ball is drawn randomly from a jar that contains 5 red balls, 7 white balls, and 4 yellow balls.…

A: From the given information, there are 5 red balls, 7 red balls and 4 yellow balls. The total number…

Q: Let D = {d₁.d₂.d3} and F = {f₁.f2.f3} be bases for a vector space V, and suppose f₁ = 5d₁-d₂ +6d3.…

A:

Q: Let V be the vector space of real 2 × 2 matrices with inner product  (A|B) = tr(B tA).  Let U be the…

A:

Q: 2 x 2 matrices and W be the subspace W X = Y Let V be the vector space of symmetric = = -2 5 (13) 0…

A: Given: Let  be the vector space of symmetric  matrices and  be the subspace(a): In part (a) we have…

Q: Let S = {a+bx+cx²: = {V1, V2, V3} and T a, b, c = R}, where v₁ = 1 + x + 3x², v₂ = −2 − 2x – 7x², v3…

A:

Q: If A is m X n, prove that every vector in null(A) is orthogonal to every vector in row(A).

A: Let A be an m×n matrix. We want to prove that every vector in the null space of A is orthogonal to…

Q: What is the length of EF? y 4 8 units 10- 7 units 4 2 units -10 -8 -6 4 10x -2- 6 units -4- -6 -8

A: solution of the given problem is below ...

Q: Let V be set of all ordered pairs of real numbers and consider the following addition and scalar…

A: Given: u=(0,4) v=(1,-3) k=2

Q: a non-zero vector in the column space of A. A = -16 -4 0 -16 20 -20 -4 5 -5 20 0 25

A:

Q: Let V be the vector space of real 2 × 2 matrices with inner product    (A|B) = tr(BtA).    Let U…

A: Let A∈skewsymm2 and B∈symm2 Then  A,B=trBtA=trAtB , since A,B=B,A=tr-AB=-trBA=-trBtA ,since B∈symm2…

Q: xux-yuy+y2u=y2x find the general solution

A:

Q: (iii) If u = 461 + 362 and [u]c = then the value of r is and the value of S is (iv) If V = 3c1 + 2c2…

A: Given that  b1 = 10c1 +3c2,  b2 =27c1 +8c2 c1 =-8b1 +3b2,   c2=27b1 -10b2

Q: Let B and C be bases for a vector space V and let the change-of-coordinates 3 . If [v]c = 7 what is…

A: Given that B and C be bases for a vector space V and let the change of coordinates matrix from B to…

Q: Let W be the set of all vectors C2 C3 CA such that 1-2x2 = 423 and 2x1 = 3 + 3x4 Determine if W is a…

A: Given that W has vectors of form               We have to check that W form a vector space or not…

Q: b) Then find the projection of b=(3,1,-3,4) onto the column space of 0 1 A 1 1 1 2

A:

Q: Let B = {bj, b2} and C = {c],c2} be bases for a vector space V, and suppose bj 3 бс1 — 2с, and bz —…

A:

Q: Consider polynomials with degree less than or equal to n. We can think that each polynomial…

A: First find map on polynomial then equivalent to tuples

Q: Let B= = : {b₁, b₂} and C = {c₁, c₂} be bases for a vector space V, and suppose V = b₁ b2 = [v] =…

A:

Q: Let Uj and U2 be vector subspaces of a vector space V over the real numbers. Suppose that v1, V2, V3…

A:

Q: Which of the following sets are closed under scalar multiplication? (i) The set of all matrices in…

A:

Q: Let D = (d₁d2₂ d3) and F = {f₁.f2.13) be bases for a vector space V, and suppose f₁ =3d₁-d₂ +5d3, f₂…

A:

Q: Consider the following vectors in R³: V1 = 1 Let V be the subspace spanned by {w₁, W2, W3}. Find a…

A: be the subspace spanned by

Q: Consider the three vectors V1 = (1,0,0, –1), v2 = (0, 1, 1, 0), V3 = (1, 1, 1, 1). Let W be the…

A: Given that W be a subspace of R4 spanned by the vectors. v1=(1,0,0,-1), v2=(0,1,1,0), v3=(1,1,1,1)…

Q: For the following let V = P2(R), the real vector space of polynomials with degree at most 2 and W =…

A: Firstly in part (a) we find the matrices corresponding to the maps T and Inverse of T. In part (b)…

Q: 5. Let V be a vector space over R, and suppose that v1, v2, v3, V4 are linearly independent vectors…

A:

Q: w that the space V = {(x1, x2, x3) ∈F3

A: To prove that the space V=(x1, x2, x3)∈F3|x1+2x2+2x3=0 forms a vector space.

Q: C: Let A = (v₁,...,04) be any list of 4 vectors in the vector space V. Let B = (v₁-V2, V₁-V3, V1-V4,…

A:

Q: Show that if V is a real vector space and av=0, then a=0 or v=0

A:

Q: Let W be the vector space of 3x3 symmetric matrices, A E W. Then, which of the following is true? a)…

A:

Q: Let C be the set of complex numbers and consider the vector space C4 with the usual vector addition…

A:

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.