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# Vector Calculus

### Vecor Spaces and Linear Combinations

Let V be the space of all functions from R to R. It was stated in the discussion session that this is a vector space over R Let F be a field, V a vector space over F, and v1,...,vk vectors in V. Prove that the set Span({v1, ..., vk}) is closed under scalar multiplication. 1. Label the following statements as true or false

### Vertex of a Parabola

A football is thrown, by a quarterback, to a receiver 40 yards away. The formula y = -0.025x^2 + x + 5 models the football's height above the ground, y, in feet, when it is x yards from the quarterback. How many yards from the quarterback does the football reach its greatest height? What is that height? keywords : find,

### Vector Spaces : Basis, Codimensions and Linear Independence

Let W= {&#945;0 + &#945;1x + &#945;2x2 +......+ &#945;n-1xn-1 ? F[x] | &#945;0 + &#945;1 + ..... + &#945;n-1 = 0}. Find a basis for W and prove that it's a basis. See attached file for full problem description.

### Vector spaces and proof of theorem

Let V be a vector space over an infinite field F. Prove that V cannot be written as the set-theoretic union of a finite number of proper subspaces. I know that I should use proof by contradiction to prove this statement, but need help from there.

### Vectors : Addition, Speed and Direction

Find |a|, a + b, a ? b, 2a, and 3a + 4b. 18. a=21?3j. bi+5j 19. a=3?2k, b=i?j+k 31. A woman walks due west on the deck of a ship at 3 mi/h. The ship is moving north at a speed of 22 mi/h. Find the speed and direction of the woman relative to the surface of the water.

### Coding Theory : Linear Code, Coset Leaders and Vector Weight

Let x+C be a coset, and assume x+c, x+e have weight less or equal than t.... See attached file for full problem description.

### Vectors : Work Done

Determine the work done by a force of F Newtons acting at a point A on a body, when A is displaced to a point B, the coordinates of A and B being (2,5,-3) and (2,-3,0) metres respectively, and when F=2i-5j+4k Newtons. Please use a diagram to illustrate your answer.

### Vectors : Force and Moment

A force of magnitude 3 units act at the point with coordinates (1,2,3). The force is applied in the direction of the vector 3i-1j+4k. find the moment of the force about O (the origin). What is the moment of the force about the point with coordinates (1,2,3). please use diagram to illustrate your answer.

### Example of a quadratic model

QUADRATIC MODELING: You will need to locate data that can be modeled using a quadratic function. Keep in mind that good candidates for quadratic models have data that both increases and decreases. Once again, I encourage you to use either online or print resources, and I would also refer you to the textbook website which has

### Vector Spaces, Basis and Closest Vector

1. Let S be a subset of R described as follows: S {(x,y,z) :x+y+z = 0} (a) Show that S is a vector space. (b) Calculate a basis of S and compute it dimension. (c) Find the vector in S which is closest to the vector (1,3, ?5) in R3.

### Inner Space Proof

Question with topic of orthogonal in inner product. See attached file for full problem description.

### Vectors, Basis, Row Space, Column Space and Null Space

1. Which of the following sets of vectors are bases and why are they bases for P2 A) 1-3x+2x^2, 1+x+4x^2, 1-7x B) 4+6x+x^2, -1+4x+2x^2, 5+2x-x^2 C) 1+x+x^2, x+x^2, x^2 2. In each part use the information in the table to find the dimension of the row-space, column-space and null-space of A and the null space

### Vectors and Gradients : Direction of Most Rapid Increase; Critical Points

1. Given f(x,y,z)=x^2y^3z^6, in what direction is F(x,y,z) increasing most rapidly at the point P(1,-1,1). What is the rate of increase? 2. Locate and classify the critical points of the function h(x,y) = x^2 -4x+4xy+y^2-16y.

### Parallel and Perpendicular Vectors and Work

1 Given a = 9i - 5j and b = 7i-4j, express i and j in terms of a and b 2 Given a=<4,5,-3> and b =<4,-2,2> determine whether a and b are parrallel, perpendicular, or neither. 3 Given F = 4i -2k;..... P(0,1,0) and Q(4,0,1) find the work W done by the force (F)moving a particle in a straight line from P to Q. 4 Given a

### Horizontal Component of a Vector

Find the horizontal component of the vector v=3971 ft at 142.0 degrees.

### Given vectors v= 150 mi at 175.0 degree and vector w= 270 mi at 215.0 degree. Find sum of v and w

Given vectors v= 150 mi at 175.0 degree and vector w= 270 mi at 215.0 degree. Find sum of v and w

### Laplace Operators and Gradient Vectors

Let f(z)=u+iv be an analytic function, phi(u,v) any function with second order partial derivatives and g(u,v) any function with first order partial derivatives. a) Let L_x,y be the Laplace operator in x,y coordinates and L_u,v be the Laplace operator in u,v coordinates. Show that L_x,y(phi o f)=L_u,v |f'(z)|^2 b)Let G_u,v be t

### Vector Spaces, Invertible Linear Operator and Change of Basis

Let be a basis of a vector space over and let be an invertible linear operator on . Then is also a basis of Show that , Please see the attached file for the fully formatted problems.

### Nonzero Vector Projections

Let U and a be nonzero vectors Show explicitly that the the angle between: Proj a U = Projection of vector U along vector a and The vector U - Proj a U (vector component of U orthognal to a) is a right angle (i.e show that the dot product of these two vectors is 0)

### Solving Vector Functions

See attached file for full problem description. Let W(s,t) = F(u(s,t),v(s,t)) ....

### Vector Spaces, Mappings and Kernels

Let V1 and V2 be finite dimensional vector space over K and let SєL(v1, V2). Prove that there exists T є L(V2, V1) such that S =STS and T= TST. See the attached file.

### Vector Calculus : Deriving the Law of Sines and Volume

Please see the attached file for the fully formatted problems.

### Equation of a Plane

Find an equation of the plane through the point (-2, -4, 0) and perpendicular to the vector (-5, -5, 0).

### Prove that a tree with Delta(T)=k (Delta means maximum degree) has at least k vertices of degree 1.

I don't understand how you count the degree of the vertices. (See attached file for full problem description) --- 2.- Prove that a tree with Delta(T)=k ( Delta means maximum degree) has at least k vertices of degree 1. Proof. We prove it by contradiction. Suppose that and there are s vertices of degree 1, where s<k.

### Vector Functions : Laplacians, Gradients and Divergence Theorem

Please see the attached file for the fully formatted problems.

### Vector Functions : Curls, Divergence and Cross Products

Please see the attached file for the fully formatted problems.

### Writing Equations from Word Problems

Two planes left an airport at noon. one flew east at a certain speed and the other flew west at twice the speed. The planes were 2700 mi. apart in 3 hours. how fast was each plane flying? Write an equation and solve.