Prove the theorem cyclically given Let W,..., W be subspaces of a vector space V. We...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Prove the theorem cyclically given Let W,..., W be subspaces of a vector space V. We say that they are independent if, whenever w W,..., WkE Wk with w + +wk = 0, then w = ... = Wk = 0. In this case, the sum W+...+W is called a direct sum and denoted by WD...OWk. If k = 2 and W W = V, then the subspaces W and W are called complementary subspaces of V. Note: For every w E W 1i k, such that w = w + Wk there exists a unique collection of vectors wi Wi, + Wk. Let V be finite dimensional. Let W,..., Wk be subspaces of V. Then the following are equivalent: (i) W,..., We are independent. (ii) W; ^ (W + + Wi-1 + Wi+1+ + Wk)= {0} for every i (iii) If B, is a basis of W, then B U... UB is a basis of W + ... 1,..., k. + Wk. Prove the theorem cyclically given Let W,..., W be subspaces of a vector space V. We say that they are independent if, whenever w W,..., WkE Wk with w + +wk = 0, then w = ... = Wk = 0. In this case, the sum W+...+W is called a direct sum and denoted by WD...OWk. If k = 2 and W W = V, then the subspaces W and W are called complementary subspaces of V. Note: For every w E W 1i k, such that w = w + Wk there exists a unique collection of vectors wi Wi, + Wk. Let V be finite dimensional. Let W,..., Wk be subspaces of V. Then the following are equivalent: (i) W,..., We are independent. (ii) W; ^ (W + + Wi-1 + Wi+1+ + Wk)= {0} for every i (iii) If B, is a basis of W, then B U... UB is a basis of W + ... 1,..., k. + Wk.
Expert Answer:
Related Book For
Elementary Linear Algebra with Applications
ISBN: 978-0132296540
9th edition
Authors: Bernard Kolman, David Hill
Posted Date:
Students also viewed these mathematics questions
-
Let W1 and W2 be subspaces of a vector space V with W1 ( W2 = {0}. Let W1 + W2 be as defined in Exercise 34. Suppose that V = W1 + W2. Prove that every vector in V can be uniquely written as w1 + w2,...
-
Let W, Z be complementary subspaces of a vector space V, as in Exercise 2.2.24. Let V/W denote the quotient vector space, as defined in Exercise 2.2.29. Show that the map L:Z V/W that maps L[z] =...
-
Hughes Company has a credit balance of $5,000 in its Allowance for Doubtful Accounts before any adjustments are made at the end of the year. Based on review and aging of its accounts receivable at...
-
How are trusts distinct from corporations?
-
What are the two basic systems of cost accounting, and under what conditions may each be used advantageously?
-
Consider the automobile gasoline mileage data in Table B.3. a. Build a linear regression model relating gasoline mileage \(y\) to engine displacement \(x_{1}\) and the type of transmission...
-
Suppose that for health reasons, the government of the nation of Ironia wants to increase the amount of broccoli citizens consume. Which of the following policies could be used to achieve the goal?...
-
K Use the piecewise-defined function to find the following values for f(x). 2-4x if x2 2x if 2
-
Consider the following income tax footnote information for Oracle for the fiscal year ended May 31, 2019 (fiscal year 2019). The following is a geographical breakdown of income before the provision...
-
A-Consider a plane wall of thickness L, the left side of the wall is maintained at Ts, and the right side with Ts. If the steady state conduction without heat generation, obtain a relation for the...
-
Royal Dutch/Shell Group discusses here its profit margins in the Chemical unit. . . . earnings in 2003 were $185 million lower. Sales volumes, including traded products, increased by 19% from a year...
-
XYZ Ltd processes and purifies a basic chemical which is then broken down by reaction to give three separate products. Explain the approaches to joint cost allocation using the following information:...
-
The following features are advertised by SAGE as being available within its job costing software module: l Define and track all cost elements of every job. With up to 10 analysis codes you can...
-
This extract is taken from a case study of a company making windows and side shutters. The company has started using a product from Best Software, a company in the Sage group. Our office is so much...
-
In a continuous flow process, the following information was collected in relation to production during the month of May: Opening work-in-progress was valued at cost of 42,000. Costs incurred during...
-
Suppose an insurance firm has three groups of insured people who have claim distribution of the form: 3000 of claims follow a N(u = 110,, o = 100) distribution , 2500 of claims folow a N(u = 120,, o...
-
Compare and contrast debt financing and equity financing as ways of starting a new business. Does one have an overall advantage over the other? What situation is more favorable to the use of debt...
-
Prove: If A and B are n n diagonal matrices, then AB = BA.
-
Suppose that S = [v1, v2, v3} is a linearly dependent set of vectors in a vector space V. Is T = {w1, w2, w3}, where w1 = v1, w2 = v1, + v3, w3 = v1, + v2 + v3, linearly dependent or linearly...
-
Prove: (a) Every matrix is row equivalent to itself. (b) If B is row equivalent to A, then A is row equivalent to B. (c) If C is row equivalent to B and B is row equivalent to A, then C is row...
-
For the particular case of hard spheres, the pressure in the virial equation of state is determined by evaluating the pair correlation function at contact. Write the pair correlation function as...
-
Use a virial expansion approach to determine the first few nontrivial order contributions to the pair correlation function \(g(r)\) in \(d\) dimensions. Show that the pair correlation function is of...
-
Show that, in the case of a degenerate gas of fermions \(\left(T \ll T_{F} ight)\), the correlation function \(g(r)\), for \(r \gg \hbar / p_{F}\), reduces to the expression \[g(r)-1=-\frac{3(m k...
Study smarter with the SolutionInn App