### Denote the vector space of all nxn matrices

The material is from ABSTRACT VECTOR SPACE. Please kindly show each step of your solution.

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The material is from ABSTRACT VECTOR SPACE. Please kindly show each step of your solution.

Choose two bases of V3(R) they should have no vectors in common and neither of them should contain multiples of the standard basis vectors e1, e2, e3. a) Prove that they are indeed bases of V3(R) b) Let one of your bases be A and the other B. Illustrate the steps of the Steinitz replacement theorem by converting B into A ste

V3(R) represents the set of vectors in 3D space. What kind of geometrical objects are represented by the various subspaces of V3(R)? i.e A 1D subspace S with basis { (0, 1, 0)Transpose} represents the set of vectors parallel to the y-axis, so the set of points with position vectors in s is the y-axis itself. You need only

Show that the following set S of vectors is linearly independent in V4(R) 2 1 3 0 -1 -2 1 2 1 -1 2 -1 (The columns should be in brackets and separated by a comma.)

Which one of these is correct and why? If T is a tree with m vertices, how many edges does T have? ans: m-1 or If T is a tree with n vertices, how many edges does T have? ans: n(n-1)/2

Let n be a positive integer. Let A be an element of the vector space Mat(n,n,F), which has dimension n2 over F. Show that the span of the infinite set of matrices span(In, A, A2, A3, ...) has dimension not exceeding n over F. Defn of the linear space Mat(n,n,F): The set of all n-by-n matrices with entries in F. Mat(n,n,F )

Show |u+v|^2+|u-v|^2=2|u|+2|v|^2

A)Find a unit vector in the direction of u=<3,2,5> b) Find magnitude U for u=a+b where a =-3i+j & b=3i-5j

Please see the attached file regarding specifics. Thank you so much for your help.

Consider the vector space R^2 with the norm ║(x,y)║ = │x │+│y │ Show that the set U = { u element of R^2 : 0< ║u║ < 1} is an open set in this normed vector space.

Show that Cn[a,b] is a subspace of C[a,b].

Let A = (0, 1) and B = (3, 2) be points on a plane. What is the length of the shortest path from A to the x-axis to B? Find where the path should touch the x-axis for this minimum to be attained and argue why it is the minimum.

I need the following in C++: A certain professor has a file containing a table of student grades, where the first line of the file contains the number of students and the number of scores in the table; each row of the table represents the exam scores of a given student and each column represents the scores on a given exam. Th

I need the following in C++. The output needs to be in a table format similar to the following sample: A demographic study of the metropolitan area around Dogpatch divided it into three regions (urban, suburban, and exurban) and published the following table showing the annual migration from one region to another (the number

Show that (A x B).(C x D) = (A.C)(B.D) - (A.D)(B.C) Show that [A x (B x C)].B = (A x B).(A x B)

15. (a) Use a counterexample to show that (A x B) x C is not necesarilly equal to A x (B x C) toAx(x). (b) Prove that A x (B x C) = (A.C)B ? (A. B)C. Can von give a geometric interpretation of this equation?

Please view the attached file for the full solution. What is presented below has many missing parts as the full question could not be copied properly. Let F be the field of real numbers and let V be the set of all sequences: ( a_1, a_2, ... , a_n, ... ), a_i belongs to F, where equality, addition and scalar multiplicat

Let F be the field of real numbers and let V be the set of all sequences ( a_1, a_2, ... , a_n, ... ), a_i belongs to F, where equality, addition and scalar multiplication are defined component wise. Prove that V is a vector space over F. See attached file for full problem description.

Find the vertices of the ellipse Xsquare/4 +Ysquare/25=1 Find equations for the asymptotes of the hyperbola X square/9 - Y square/81 = 1 Change the polar coordinates of (5,pi) to rectangular coordinates.

In a vector space show that a( v - w ) = av - a w. See attached file for full problem description.

Find the equation of a plane through the origin and perpendicular to: x-y+z=5 and 2x+y-2z=7

Find the equation of the tangent to the surface at the indicated point: x = u^(2), y = v^(2), z = u + v ; (0, 2, 0) .

1. Let W = {(a, b ,2a - 3b, -a + 2b)} whre a and b are real numbers. (a) In what Euclidean space does this subset reside? Explain your answer. (b) Show that W is a subspace by showing that it satisfies the closure properties. (c) Show that W is a subspace by describing W as the span of a set of vectors. (d) Explain why this

Please help me with 2 attached problems. In problem 6 I can imagine the space and guess which vectors are orthogonal to (1,-1,0) like (0,0,1) for example. However, I am not sure of my approach when I do it by hand. I do not know how to approach problem 12. I am doing something completely wrong and especially need help with

Find k(t) for y = 1/x. ^ P.S. The curvature of a curve is k = dT/ ds, where T is the unit tangent vector. ^ ^ And k(t) = T'(t)/ r'(t)

^ ^ ^ ^ ^ For v = -4 j , a = 2 i + 3 j find a (subscript)T and a (subscript)N.

2. Use Theorem 5.2.1 to determine which of the following are subspaces of M22. Thm 5.2.1: If W is a set of one or more vectors from a vector space V, then W is a subspace of V if and only if the following conditions hold. (a) If u and v are vectors in W, then u + v is in W. (b) If k is any scalar and u is any vector in W,

Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and thaw the velocity and acceleration vectors for the specified value of t. 3, r(t)= <t^2 ? 1, t>, t=1 4. r(t)=<2?t, 4 sqrt t), t=1 5. r(t)=e^t i+e^-t j, t=0

5. A particle P is acted on by three forces F1, F2 and F3, where F1 = (2i - 5j)N and F2 = (4i - 4j). Given that P is in equilibrium, a. Find F3 in terms of i and j The force F3 is now removed and P moves under the action of F1 and F2 alone. b. Find to 3 s.f. the magnitude of the resultant force acting on P. c. Find, in

^ Find the domain of r(t) = <2e^(-t), sin^(-1) t, ln (1 - t) >; t(subscript 0) = 0