Purchase Solution

Subsets, Projection Maps, Basis and Direct Sums

Not what you're looking for?

Ask Custom Question

Let n >= 1. Define the subsets U and W in V = F^n as follows:
U = {(x_1, . . . , x_n) : x_1 + . . . + x_n = 0} W = {(x_1, . . . , x_n) : x_1 = . . . = x_n}
a) Prove that U and V are subspaces of V .

b) Prove that V = is the "direct sum" of U and W.

c) Let (v_1, . . . v_n) = ((1, 0, . . . , 0), (0, 1, . . . , 0), . . . , (0, . . . , 0, 1)) be the standard basis of V, and let
P_U,W and P_W,U denote the projection maps. Find P_U,W(vi) and P_W,U(vi).

Purchase this Solution
Solution Summary

Subsets, Projection Maps, Basis and Direct Sums are investigated. The solution is detailed and well presented.

Purchase this Solution
Free BrainMass Quizzes
Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Probability Quiz

Some questions on probability

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

View More Free Quizzes
Related BrainMass Solutions

  • Nonzero subspace

    The question illustrates an important part of the definition for when the direct sum is applied to more than two subsets.

  • K-Mapping Design Problems

    Fill in the truth table and use k-maps to calculate the minimal sums for all of the output bits. Please see attached for the solution to the given problems.

  • World projections

    When you change the projection, what happens to their shape and areas? Describe the differences you see. World projections are examined and discussed in the solution.

  • Stereographic Projection, Riemann Sphere and Mapping

    Given this mapping the southern hemisphere maps inside the circle of radius 1, the equator stays where it is and the north hemisphere maps onto the region outside the unit circle. Note that the North Pole maps onto "infinity".

  • Diagonalizable Matrices, Image, Kernels and Direct Sums

    103734 Diagonalizable Matrices, Image, Kernels and Direct Sums Prove that if T Є L(V) is diagonalizable then V = im(T) + ker(T) (+ = direct sum) (Hint: Use a basis of eigenvectors.

  • discrete topology

    Show that, if Y has the discrete topology and if p: X x Y --> X is the projection onto the first factor, then p is a covering map. In the discrete topology all subsets are open. In particular, all sets that contain a single point are open.

  • Important information about Homology group

    All continuous maps from connected spaces C to either the rationals or Z (a countable discrete set) are constant, as the only connected subsets are the singletons, in both cases.

  • Equivalence relations, surjective maps, partitions and fibers.

    Equivalence relations, surjective maps, partitions and fibers are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

  • Show that the product of two step functions is a step function

    Now, suppose f, g are step functions on subsets A, B of the real line, respectively; then they have the representations f = sum_{i=1}^n a_i chi_{A_i} g = sum_{i=1}^m b_i chi_{B_i} as finite sums, where the {A_i} and {B_i} are pairwise

View More Related Solutions