1. a. b. Let V be the set of real-valued functions that are defined at each x in the interval (-∞, ∞). If ƒ = f(x) and g = g(x) are two functions in V and if k is any scalar, then define the operations of addition and scalar multiplication by (ƒ+g) = f(x) + g(x) kƒ = kf(x) Verify the Vector Space Axioms for the given set of vectors. Let V consist of the form ū = (U₁, U2, ..., Un, ...) in which 1, U1, U2, ..., Un, ... is an infinite sequence of real numbers. Define two infinite sequences to be equal if their corresponding components are equal, and define addition and scalar multiplication componentwise by u + v = (u₁, U2, ..., Un, ...) + (V1, V2, ..., Un, ...) 1, (u₁ + v₁, U2 + V2, ..., Un + Un, ...) = ku = (ku₁, ku2, kun, ...). Prove that the given set is a vector space.

Q: Let L be the line in R2 through the points [:] } Find a linear functional f and a real number d such…

A: First, we will find the equation of a line given two points.

Q: Which of the following mappings A: R^2->R is a linear functional? (a) (a,b) -> lal + lbl. (b)…

A: Step 1:A linear functional is a linear map from a vector space to its underlying field. Formally,…

Q: 12. Show that the set of all continuous function on the interval [-1,1] is a vector space under…

A:

Q: Show that λis a root of char(x) if and only if there exosts a non-zero vector v in V with L(v)=λv

A: Assume that λ is a root of charL(x) then | λIdv–L | = 0. It is known that, the system of Bx = 0 has…

Q: Let v1 =[1,4,-2] , v2 =[-2,-3,7] , b =[4,1,h] FIND all value(s) h such that b is in span{v1 , v2 }

A:

Q: The linear operator L : R² R² given by L (²) 6x - a) (55x + y) = -1 Which of the following gives a…

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: Prove that V is a vector space with the operations

A:

Q: 10. Show that V = R² with the standard scalar multiplication, but addition defined by (x, , y.) +…

A: We have to show that the set V with the addition operation x1, y1+x2, y2=x13+x233, y13+y233 and…

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

A:

Q: Let φ1(x) = 1, φ2(x) = x/|x|, φ3(x) = x^2 − 1/3 for −1 ≤x ≤1. Show that (φ1,φ2,φ3) form an…

A:

Q: Please note that a bold letter indicates a vector and that this problem involves a vector function…

A: Function is given as,

Q: Z is the set of integers, R is the set of real numbers. 1) Functions f, g, h are defined as f: R ->…

A:

Q: Let P3 is the vector space of polynomials of the degrees3,Consider the polynomials…

A: Given :  P3 is the vector space of polynomials of degree less than or equal to 3. To prove :  We…

Q: Suppose f is the function defined by f(x) = Ax, where a. The domain of f is b. The codomain of f is…

A: Dear student, as per our guidelines we can answer only first three subqustions. So kindly resubmit…

Q: 7. Find if the following expressions result in a scalar or vector quantity or they are meaningless.…

A:

Q: one example of a 4-dimensional vector space that is not R 4

A: