Consider vector space V over field F. Consider set of all linear mappings from vector space V to vector space V, i.e. set L(V). Linear mappings from L(V) can be naturally added and multiplied by a scalar from F. Select all true statements. Set L(V) with given operations is a vector space. V = ker A U A(V) for every linear mapping A from L(V), because dim V = nul(A) + rk(A). Multiplication of linear mapping from L(V) by a scalar from F is again a linear mapping from L(V). Composition of two linear mappings from L(V) is again a linear mapping from L(V).

Q: 2) Let V₁, V2, W be vector spaces over F. Show that the set Bil(V₁ × V₂, W) of bilinear maps is a…

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

A: Vector space: A non-empty set V closed with two binary operations called addition and scalar…

Q: Consider the vector space P, of polynomials with real coefficients of degree at most two together…

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

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

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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: With T defined by T(x) = Ax, find a vector x whose image under T is b, and determine whether x is…

A: The given matrix  and the vector  and  is defined by .The aim is to find the vector  whose image…

Q: 3. Prove that, for a given n, all n-dimensional vector spaces with real scalars are isomorphic.

A: A vector space V will be isomorphic to other vector space W if there exists a linear transformation…

Q: Define a function f : C -> C by f(x+iy) = (x+2y) + i(3x+4y) for x,y in R. Show that f is additive…

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Q: Show that the set of all second order polynomials, f(x) = ax² + bx+c, is a vector space.

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Q: Question Consider the vector space P, consisting of all 2 polynomials of degree at most two together…

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Q: . Let V be a vector space, and {v1, ··· , vn} be a set of vectors that spans V . Show that the set…

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Q: Zero is the only vector in an inner product space V that is orthogonal to itself. O True False

A: The statement is true. We give the justification in the next step.

Q: Determine whether the set, together with the indicated operations, is a vector space. If it is not,…

A: Given that S(say) is the set of all quadratic functions , whose satisfies the point (0,7). Then we…

Q: Indicate which of the following is a vector space over R. (Justify your answer)

A: The objective of the question is to determine whether the set B = {(x, y, z); x = y - z - 1} forms a…

Q: Prove that R is an infinite-dimensional vector space.

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Q: Let U be a subspace of a finite-dimensional Prove that dim(U) = dim(V) if and only if U = V. vector…

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Q: Define vector space and determine whether the set of all positive real numbers u with the operations…

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Q: Determine if each of the following sets with the indicated operations of addition and scalar…

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Q: Let S be the set of all ordered pairs of real numbers (x1, x2) equipped with the following…

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Q: 3. [6 points] Consider the vector space V = P, over the field of real numbers F = R. Is the vector 4…

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Q: Let Riz be a linear space of all polynomials p(x) = ar²+bx+c (a,b,c & R) whose degree is no more…

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Q: 2. Prove that every element in a vector space has a unique additive inverse.

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Q: Suppose {v₁, ..., vr} is linearly independent but does not generate the vector space V. Show there…

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Q: ive an example of a space is not vector space

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Q: All solutions of the differential equations y" - 5y' + 6y = 0

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Q: 5. Prove that (-1)v=-v for every vector v in a vector space.

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Q: Let V = {(x, 6) | x E R}. You are given that (V,O, O) is a vector space with (x1, 6) O (x2, 6) =…

A: The objective of the question is evaluate the zero element and inverse element.

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: Show that the set of real polynomials of n – degree (Pn) is a vector space with traditional addition…

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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: Let E and F be two vector spaces with the same finite dimension dim(E) = dim(F) For every linear map…

A: Follow the steps.

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.