3. Let V denote the vector space of all functions f: R" → R, equipped with addition + : V × V → V defined via (f+g)(x) = f(x) + g(x), x = R", and scalar multiplication : Rx V → V defined via (Af)(x) = \ƒ(x), xER". Now let W= {f: R → R: f(x) = ax + b for some a, b = R}, i.e. the space of all linear functions R" → R. (a) Show that W is a subspace of V. (You may assume that V is a vector space). (b) Find a basis for W. You should prove that it is indeed a basis.

Q: Define two vector functions F(t) = 8 sin(t)i + 6 cos(t)j + (t – 2)k ü(t) = 6 sin(t)i + 8cos(t)j + (t…

A: We have to find the dot product of r(t) and u(t)

Q: 3. Calculate the directional derivative of the function f : R³ → R, defined by f(x, y, z) = y²+xz at…

A: Step 1: Step 2: Step 3: Step 4:

Q: Suppose f(x, y) = (a) ▼ f(x, y) = (b) ▼ f(-0.7, 7) = tan(x) + y and u is the unit vector in the…

A: Given that f(x,y)=tan(x)+y  and u is the unit vector in the direction of <2,3>now, ∇f(x,y)=fx,…

Q: 1. Let f: R" → R be the function f(x) = cx+b for some fixed vector c ER" and scalar b. Show that f…

A:

Q: , Let C" (R) be the vector space of "smooth" functions, i.e., real-valued functions f(x) in the…

A:

Q: Find the point on the line T = (-2, 2, 3) + t(4, 1, 1) that presents the shortest distance to the…

A:

Q: xy² (0) (Use syntax like [3,-4,5] for vectors.) Let F (x, y, z) = xz5 What is the value of curl F at…

A: The vector F(x, y, z) is given. We have to find the value of curl F at (3, 2,1) .

Q: Find the domain of the vector functions, r(t), listed below. You may use "-INF" for -∞o and use…

A: According to bartleby guidelines we supposed to do the first one of multiple questions kindly repost…

Q: Let C[0, 1] be the vector space of continuous functions that are defined on the interval [0, 1].…

A:

Q: a) Show that if u, v, and w are differentiable vector functions of t, then d du V X w +u - dt dv dw…

A: According to our guideline we are supposed to answer only one asked question.kindly repost other…

Q: consider the vector-valued function r(t) = a) State the domain of r(t) b) Identify where r(t) is…

A:

Q: (a) Find the best approximation to ✅ by a vector of W. (b) Write as the sum of a vector in W and a…

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

Q: Suppose f(x, y) = √tan(x) + y and u is the unit vector in the direction of (1, 1). Then, (a) ▼ ƒ (x,…

A:

Q: f(2) = 2z2 – iz - be a complex valued function defined on a complex plane. Writing z = x + iy, find…

A:

Q: 3. Let V denote the vector space of all functions f: Rn R, equipped with addition + : V × V → V…

A: