Let V be the set of ordered pairs (x, y) of real numbers and let F be the field of real numbers. Define (x. y;)+ (x,. Va)= (x + x,. 0) c (x. i)=(cx.0). Examine whether V is a vector space or not ? If not, list the properties which fail to hold.

Q: Use proof by contradiction to prove the statement.In a vector space, the zero vector is unique.

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Q: Verify that this is a vector set check each property(assoctive, etc): Y = (yo, y#) (xo + yo + 1, x#…

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Q: (a) A: va+v. (b) A: (av). (c) A: √ av.

A: “Since you have posted a question with multiple sub parts, we will provide the solution only to the…

Q: A subspace always contains the zero vector.

A: True Because of subspace is a nonempty set.

Q: The scalar product 0v is 0.

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Q: _et a₁ = -1 a₂= -7 -7 3 and b = 3 -15 h For what value(s) of his b in the plane spanned by a, and a

A: If b is in the plane spanned by a1 and a2 then b can be written as linear combination of  a1 and a2…

Q: Determine if the set W of vectors (X1 , X2) in R such that x1² + x2 = 0 is a subspace of R?.

A: We have to determine whether W is a subspace of R^2 or not.

Q: Determine the dimension of the vector space given by -a V = | a,b E R a b

A: Given V = -aa0, -b0b | a, b ∈ℝ

Q: Let u = (5,-5, -7) and 7 = (-6, 1, 2). Compute the following: úxv= v X ū= úxu 3 (u xv) = ū × (30) =…

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Q: Consider the set V = R − {2}, that is, the set of all real numbers except 2. Define “addition” as…

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Q: c. Show that V, with respect to these operations of addition and scalar multiplication, is not a…

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Q: Let a, 1 1 -7 -7 3 and b = 3 15 h For what value(s) of h is b in the plane spanned by a, and a₂?

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Q: Determine whether the following set of vectors are linearly independent

A: Step 1: Step 2: Step 3: Step 4:

Q: = ---- -2 8 span of v₁ and v₂? Let V₁ = and y = -5 -3 For what value(s) of h is the vector y in the

A: The given vectors  and .The objective is the find the value of  for which the vector  in the span of…

Q: Let V be the set of all ordered pairs of real numbers (x1, u) with u > 0. Consider the following…

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Q: Describe the zero vector (the additive identity) of the vector space. P3 x + ]+² +

A: We have to describe the zero vector (the additive identity ) of the vector space P3.

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: Let A = [[3,1,1,2,2],[-3,-2,4,2,2],[-5,5,4,-1,-2]]  Give a nonzero vector x in the nullspace of A.

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Q: Write the interpretation of independent vectors, span, basis and dimension. Take the example of…

A: We have to give the interpretation of independent vectors, span, basis and dimension by taking the…

Q: Find a linear mapping G that maps [0, 1] × [0, 1] to the parallelogram in the xy-plane spanned by…

A: The linear map G maps  to the parallelogram in XY-plane.The parallelogram is spanned by the vectors…

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

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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.