Explain if the set below is a vector space given standard operations. The set of all...

Explain if the set below is a vector space given standard operations.

The set of all even functions defined on R with addition and scalar multiplication defined as follows:

1.) (f+g)(x) = f(x) + g(x) (addition)
2.) (cf)(x) = cf(x)

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

QUESTION 1 Vector Space Axioms Let V be a set on which two operations, called vector...

QUESTION 1 Vector Space Axioms Let V be a set on which two operations, called vector addition and vector scalar multiplication, have been defined. If u and v are in V , the sum of u and v is denoted by u + v , and if k is a scalar, the scalar multiple of u is denoted by ku . If the following axioms satisfied for all u , v and w in V and for all scalars k...

Verify all axioms that show that the set of second degree polynomials is a vector space....

Verify all axioms that show that the set of second degree polynomials is a vector space. What is the Rank? P2 = {p(x)P | p(x) = ax^2 + bx + c where a,b,c E R}

1. Show that the set of all polynomials of deg=2 is not a vector space over...

1. Show that the set of all polynomials of deg=2 is not a vector space over reals. can this be fixed, can we have a set of polynomials that is a vector space over reals? 2. Show that the set of 2x2 matrices with m_22 = 1 is not a vector space over reals. 3. Show that the set of infinitely-differentiable real functions is a a vector space under pointwise function addition, and pointwise scalar multiplication as defined in class,...

(1) Suppose that V is a vector space and that S = {u,v} is a set...

(1) Suppose that V is a vector space and that S = {u,v} is a set of two vectors in V. Let w=u+v, let x=u+2v, and letT ={w,x} (so thatT is another set of two vectors in V ). (a) Show that if S is linearly independent in V then T is also independent. (Hint: suppose that there is a linear combination of elements of T that is equal to 0. Then ....). (b) Show that if S generates V...

A basis of a vector space V is a maximal linearly independent set of vectors in...

A basis of a vector space V is a maximal linearly independent set of vectors in V . Similarly, one can view it as a minimal spanning set of vectors in V . Prove that any set S ⊆ V spanning a finite-dimensional vector space V contains a basis of V .

Define a subspace of a vector space V . Take the set of vectors in Rn...

Define a subspace of a vector space V . Take the set of vectors in Rn such that th coordinates add up to 0. I that a subspace. What about the set whose coordinates add up to 1. Explain your answers.

Let V be a finite dimensional vector space over R. If S is a set of...

Let V be a finite dimensional vector space over R. If S is a set of elements in V such that Span(S) = V , what is the relationship between S and the basis of V ?

Given the data set below, calculate the standard deviation standard deviation using the defining formula standard...

Given the data set below, calculate the standard deviation standard deviation using the defining formula standard deviation using the computing formula 8,41,24,19,15,12,47,12,33

Vector v=(9,0,2) is vector from R3 space. Consider standard inner product in R3. Let W be...

Vector v=(9,0,2) is vector from R3 space. Consider standard inner product in R3. Let W be a subspace in R3 span by u = (9,2,0) and w=(9/2,0,2). a) Does V belong to W? show explanation b) find orthonormal basis in W. Show work c) find projection of v onto W( he best approximation of v with elements of w) d) find the distance between projection and vector v

Determine if the following sets are real vector spaces with the indicated operations. (a) The set...

Determine if the following sets are real vector spaces with the indicated operations. (a) The set V of all ordered pairs (x, y) of real numbers with the addition defined by (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2 + 1) and scalar multiplication defined by α(x, y) = (αx, αy + α − 1), α ∈ R (b) The set V of all 2 × 2 real matrices X with the addition of M22 but...

Subjects