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For each of the following vector spaces, give its dimension and a basis.
(a) the set of all symmetric 4 x 4 matrices,
(b) the subspace of P3 consisting of those polynomials in P3 whose graphs pass through the origin
(c) the set of all vectors in R3 that are orthogonal to v = (1,2,-1)
Linear Spaces, Mappings and Dimensional Spaces, 1) Show that if dim X = 1 and T belongs to L(X,X), there exists k in K st Tx=kx for all x in X.
These questions are about the quadratic function which takes the form: Choose the false statements.
Find the derivatives of the following functions (Do not simplify): f(x) = 20x^(11/5) - 13x^(5/4) - 4x(-7/2)
When n is a positive integer by using mathematical induction and the formula of d/dz[f(z)g(z)]=f(z)g'(z)+f'(z)g(z) for the derivative of the product of 2 functions.
A consumer's group checked the gasoline mileage (to the nearest mile per gallon) on 25 different cars. The following gives a summary of the findings:
The gas law for a fixed mass m of an ideal gas at absolute temperature T, pressure P, and volume V is PV=mRT, where R is the gas constant. Show that
Attendance at the Fireman's Ball has been declining. Each year, the number attending is 2/3 of the previous year's attendance and the cost of food is 3/4 that of the previous year.
Write equations for vertical and horizontal lines passing through the point (-6,5) in (x,y) coordinates. A line passes through the point (x,y)= (-9,-5)and has a slope of -8. Write an equation for the line.
The plane y + z = 3 intersects the cylinder x^2 + y^2 = 5 in an ellipse. Find the parametric equations for the tangent line to this ellipse at the point (1,2,1)
Explaining how to derive the formula for the distance between two points -Prepare an essay of 1,000 words explaining how to derive the formula for the distance between two points in analytic geometry 3-space.
Determine if the given function can be extended to a continuous function at x=0. If so, approximate the extended function's value at x=0 (rounded to four decimal places if necessary).
Suppose that a "skew" product of vectors in R2 is defined by (u,v)=u1v1-u2v2. Prove that (u,v)squared >equal too (u,u)(v,v). (NOTE; This is just the reverse of the Cauchy- Schwartz inequality for the ordinary dot product.)
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