Question: Let T: V W be a linear transformation, where V and W are finite dimehsional. Show that T is onto if and only if

Let T: V → W be a linear transformation, where V and W are finite dimehsional.
Show that T is onto if and only if there exists a linear transformation S: W → V with TS = 1w A [Let {e1,..., er ,..., en} be a basis of V such that {er+1,..., en] is a basis of ker T. Use Theorem 5 §7.2 and Theorems 2 and 3, §7.1.]

Step by Step Solution

★★★★★

3.46 Rating (166 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

If TS 1 W then given w in W TSw w so T is onto Con... 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)

950-M-L-A-L-S (6582).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Prove Theorem 6.14(b). Let T : VW be a linear transformation. Then: T(-v) = - T(v) for all v in V.

Let L: V W be a linear transformation. If {v1, v2,..., vk} spans V, show that {L(v,), L(v2),..., L(vk)} spans range L.

Let T: V V be a linear transformation. Show that T2 = 1 y if and only if T is invertible and T = T-l.

In each case either prove the statement or give an example in which it is false. Throughout, let T: V W be a linear transformation where V and W are finite dimensional. (a) If V = W, then ker T im...

answer given questions showing all work (2) Let T : V > W be a linear transformation, and let S be a subspace of V. Prove that the set T(S) = {0: E W|T() = o: for some 6 E S} is a subspace of W. (9)...

Explain the following que.stions as attached below. Pages 2-5 Q1 a) Let V be a finite dimensional vector space over a field F. If S and S' are two different bases of V, then prove that the number of...

Reveal an important connection between linear independence and linear transformations and provide practice using the definition of linear dependence. Let V and W be vector spaces, let T: V W be a...

1: Let V and W be vector spaces over a field F, and let T : V W be a linear transformation. Suppose that ker(T) and im(T) are both finite dimensional, that {v1,...vn} is a basis for ker(T), and that...

Reveal an important connection between linear independence and linear transformations and provide practice using the definition of linear dependence. Let V and W be vector spaces, let T: V W be a...

:) will leave good feedback Let T : V - W be a linear transformation, and let { V1, V2, ... , Vx } be vectors in 1'. Select the True statements: O If { v1, V2, ... , Vx } spans I/ then IT(v1), T(v2),...

Paul Ward is interested in the stock of Pecunious Products, Inc. Before purchasing the stock, Ward would like your help in analyzing the data that are available to him as follows: Mr. Ward would like...

The current (year 0) price of the shares of Company XYZ is $50. There are 1 million shares outstanding. Next year (year 1)'s dividend per share is $2, which represents a 60% payout from earnings (net...

5. p chart is used when the attribute follows______ distribution.

Andreini suffered from a nerve problem that was causing him to lose the use of his hands. Dr. Beck operated on Andreinis hands, but the problem grew worse. A nurse told Andreini that Beck might have...

Show that if A , B, and C are square matrices, then det = (det A) (det B). A O C B

Use the properties of Section 3.2 to prove that laa

Theorem 3.8 assumes that all calculations for det(A) are done by exact arithmetic. As noted previously, this is usually not the case in software. Hence, computationally, the determinant may not be a...

You want to buy a car, and a local bank will lend you $30,000. The loan would be fully amortized over 4 years (48 months), and the nominal interest rate would be 6%, with interest paid monthly. What...

Question: Don't attempt if you are not sure In this problem the geometric(p) distribution is defined as the total number of trials to the first failure (the value includes the failure), where p is...

Let (X, Y ) have joint PDF f(x,y) = re -" for r > 0, y > 0 and f(r, y) =0 otherwise. (a) Compute the marginal PDFs of X and Y. (b) Determine whether X and Y are independent