Question: Let U and V be subspaces or Rn. In the case that U V = {0} we have the following dimension relation dim (U

Let U and V be subspaces or Rn. In the case that U ∩ V = {0} we have the following dimension relation
dim (U + V) = dim U + dim V
(See Exercise 18 in Section 4 of Chapter 3.) Make use of the result from Exercise 13 to prove the more general theorem
dim (U + V)= dim U + dim V - dim({/ n V)

Step by Step Solution

★★★★★

3.55 Rating (162 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

Let m dimU k dim V and W U V If dim W r 0 and v 1 v r is a basis for W then by Exe... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

949-M-L-A-E (760).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Let U and V be subspaces of a vector space W. Prove that their intersection U V is also a subspace of W.

Let U and V be subspaces of a vector space W. Define U + V = {z | z = u + v where u U and v V} Show that U + V is a subspace of W.

Let U and V be subspaces of a vector space W. If W = U V, show that V = (0).

Let U and V be subspaces of R ^ n . In the case that U V = { 0 } , we have the following dimension relation: dim ( U + V ) = dimU + dimV

15. (Dimension Theorem) Let U and V be subspaces of R" . In the case that Un V = {0} , we have the following dimension relation: dim(U + V) = dimU + dimV (See Exercise 18 in Section 3.4 0.) Make use...

Show that if U and V are subspaces of Rn and U V = {0}, then dim (U + V) = dim U + dim V

Let U and W be subspaces of V, let dim V = n and assume that dim U + dim W = n. (a) If U W= {0}. show that V = UW. (b) If U+W= V,, show that V = UW. [Hint: Theorem 5 6.4.]

Let U and W be subspaces of a finite-dimensional vector space V. Define T: U W V by T(u, w) = u - w. (a) Prove that T is a linear transformation. (b) Show that range(T) = U + W. (c) Show that...

9. Let U and W be subspaces of a vector space V. (a) If dim V = 3, dim U = dim W = 2, and U 7E W, prove that dim (U NW) 2 1. (b) Interpret the result from part (a) geometrically if V = 1R3

Let U and W be subspaces of a vector space V. (a) If dim V = 3, dim U = dim W = 2, and U W, show that dim(U W) = 1. (b) Interpret (a) geometrically if V= R3.

Why did convertible bonds perform poorly in 2008 and then perform very well in 2009? Why did convertible arbitrage strategies do poorly in 2008? We have the following information: Conversion ratio=...

Compute the Euclidean distance between the following sets of points: a. (2,5) and (8,4) b. (2, - 1, 32)and (8, 15, - 5)

20. (a) Propene is used to make alcohols and poly(propene). (i) Describe how propene can be converted into an alcohol and draw the structure of this alcohol. [2] (ii) Draw the structure of...

3. Does the economy this year seem to have an inflationary gap or a recessionary gap? (If you do not know the answer from reading the newspaper, ask your instructor.)

In each case, find the matrix of T:V W corresponding to the bases B and D, respectively, and use it to compute CD[T(v)], and hence T(v). a. T: R3 R4, T(x,y,z) = (x + z,2z,y - z, x + 2); B and D...

In each case, verify Therorem 3. Use the standard basis in Rn and [1,x,x2] in P2 a. b. c. d. S(a, b)-(a, b-2a, 3b, a+b) T(a, b, c) (atb, c+b, at, b-a), S(a, b, e, d) (a+b, c-d) T(a + bx + cx*)=(a,...

In each case, find T-1 and verify that MDB (T)-1 = MBD(T-1). (a) T: R2 R2, T(a,b) = (a + 2b, 2a + 5b); B = D = standard (b) T: R3 R3 T(a,b,c) = b + c, a + c, a + b); B = D = standard (c) T : P2 ...

Beryl's Iced Tea currently rents a bottling machine for $50,000 per year, including all maintenance expenses. It is considering purchasing a machine instead and is comparing two options:a. Purchase...

CHECK FIGURES: Prolit =$8,150 , Tolal assets =$14,250 \ Using your answor trom Exorciso 1-20, propare an incomo statement, a statement of changas an equify, and a balance sheet using tho formats...

On Dec. 31, 2018, what was the last price of an Apple (AAPL-US) call option expiring on Mar. 15, 2019 with a strike price of $140.00? O $7.90 O $21.70 O $14.20 O $11.30 On Dec. 31, 2018, what was the...