If on V = R² define the usual scalar multiplication « (x, y) = (« x,x y) , and addition (x1, Y1) + (x2, Y2) = (x1 + x2 + 1,y1 + y2 + 1) . (hint: show that ( a + B ) x = a x + ßx is not satisfied ) %3D If on V = R² define the usual addition (x1, y1) + (x2, Y2) = (x1 + x2 , Y1 + Y2) and scalar multiplication « (x, y) %3D = (x y, x x). (hint: lu = u is not satisfied )

Q: Q3) Determine whether V is a vector space with the following operations. If it is, verify each…

A: We have to solve given problem:

Q: Suppose u1,..., u, and v1, . Span {u¡, ..., u„} and K = Show that H + K = Span {uj,., up, V1, ...,…

A:

Q: If V is a vector space than show that cV=V

A:

Q: There was no information/work on if V is a vector space

A:

Q: What should k be if the vectors x=(1,1,k) y=(1,3,1) z=(k,1,3) are in the same plane in the Euclidean…

A: Three vector a1,a2,a3,b1,b2,b3 and c1,c2,c3 will be in euclidean space ℝ3 If  a1a2a3b1b2b3c1c2c3=0…

Q: Is it possible to make Z into a vector space if we redefine addition?

A: To check: the possibility to make ℤ into a vector space.

Q: State whether the following statement is true or false: A vector space may have two or more zero…

A:

Q: Show that, in an inner product space, there cannot be unit vectors u and v with <u, v > <…

A:

Q: Write down a basis for the vector space L(R", R).

A: Given vector space is the finite dimensional vector space of dimension m.

Q: (a) Give examples of three distinct vectors v1, v2, v3 in the vector space R3 such that v2 E…

A: Let v1 = {1,1,1}v2 = {2,3,4}v3 = {1,2,3} Here v2 = {2,3,4} = {1,1,1}+{1,2,3} = v1+v3

Q: Prove that M2,3 is a vector space

A: To prove that M2,3, the set of 2 by 3 matrices, is a vector space, we need to show that it satisfies…

Q: Determine if S=R (real numbers) where -1 is not included, is a vector space with the operations:…

A: Vector space

Q: Please explain why  If V� is a vector space and u� and v� are vectors in V�, then according to the…

A: V is a vector space and u and v are vectors in V.

Q: Give two examples of infinite dimension vector spaces.

A:

Q: 1) Let v be an element of a vector space V. Prove that -(-v) = v. Your proof should work in any…

A: Here a,b are element of field.

Q: Show that the vector spaces V = M2 and W = R4 are isomorphic

A: A linear transformation (or a linear map) is a function T from one vector space V to another vector…

Q: Let (V, (, )) be an inner product space and let v, w be vectors in V. If (v, w) = 7 and w = 5, then…

A:

Q: Show that the function (V1, V₂) = 2w1W2 + x1x2 - Y1Y2 +2122 defined for any vectors V₁ = (x1, y1,…

A:

Q: 20. Prove that a vector u in a vector space has only ative, -u. one neg-

A: By definition of negative of an element in a vector space, we have        u + (-u) = (-u) + u = 0 .…

Q: What is the dimension of the vector space P°?

A: The dimension of Pn is n+1

Q: 2. Let W1 and W2 be two subspaces of V, such that dimg(W1) = dimg(W2) = 2. Suppose that the only…

A: here we can use linear combination to prove that given set is linearly independent.

Q: Show that the set is a sub-vector space of space R3 and find the generator set of this sub-vector…

A: A typical vector in H = (2t, t - k, -2t + 3k) = (2t, t, -2t) + (0, -k, 3k) = t(2, 1, -2) + k(0, -1,…

Q: In Exercises 3-4, determine whether b is in the column space of A, and if so, express b as a linear…

A:

Q: Which vector in two-dimensional space has no corresponding unit vector?

A:

Q: Let L : V → W be a mapping, where V and W are vector spaces. Which of the following is NOT always…

A: Let L:V→W be a mapping, where V and W are vector spaces. We have to find the false statement of the…

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: Does the set of all functions that vanish at x = 0 and x = L form a vector space? If it does,…

A: 1) Commutativity of addition: Let f(x) and g(x) be functions in V.Then, .Therefore, addition is…

Q: Give two examples of finite dimension vector spaces.

A:

Q: Any given vector space may have distinct inner products.  True  False.

A: Given :- Any given vector space may have distinct inner products. To find :- The given statement is…

Q: Please note that f denotes a vector function defined by the set of vectors V in E (one dimensional…

A: Inverse function theorem-

Q: Give a non-zero vector in the null space of A. 4 A = -4 -5 2 4-3 -55-3 -1 1-5 -3.

A:

Q: Is the set of (infinite number of) vectors { (0,0,c) }, where c is a real number, a vector space…

A:

Q: Problem 4 Let V be the span of the following vectors F 3 V1 = U2 = -2 2 V3 = 8 6 U4 = V5 = V6 = -1 5…

A: Result :- The more vectors than the dimension of a finite dimensional vector space are always…

Q: Q3) Determine whether V is a vector space with the following operations. If it is, verify each…

A: V does not form vector space, because

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.