Question: Let T: V W be a linear transformation between finite-dimensional vector spaces and let B and C be bases for V and W, respectively.

Let T: V → W be a linear transformation between finite-dimensional vector spaces and let B and C be bases for V and W, respectively. Show that the matrix of T with respect to B and C is unique. That is, if A is a matrix such that A [v]B = [T(v)]c for all v in V, then A = [T]C←B.

Step by Step Solution

★★★★★

3.55 Rating (166 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

Let B v 1 v n and suppose that A is such that Av B Tv C for all v V Then Av i ... 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)

859-L-A-L-S (2808).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

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.

In Exercises 1-3, let T: V W be a linear transformation between finite-dimensional vector spaces V and W Let B and C be bases for V and W, respectively, and let A = [T] CB. 1. Show that nullity(T) =...

Let T : V W be a linear transformation between finite-dimensional vector spaces V and W. Let B and C be bases for V and W, respectively, and let A = [T] CB . If dim V = n and dim W = m, prove that L...

Let T : V W be a linear transformation between finite-dimensional vector spaces V and W. Let B and C be bases for V and W, respectively, and let A = [T] CB . Use the results of this section to give...

Let T : V W be a linear transformation between finite-dimensional vector spaces V and W. Let B and C be bases for V and W, respectively, and let A = [T] CB . If B and C are also bases for V and W,...

Math 416-D13, Abstract Linear Algebra, Spring 2017 Homework 6, Due Friday, March 10, 2017 Read [FIS] Sections 4.1-4.2 Problems 1. Suppose T : V W is a linear transformation between finite-dimensional...

40. Consider the following two systems of equations: 5x1 + X2 - 3x3 = 0 5x1 + X2 - 3x3 = 0 -9x1 + 2x2 + 5X3 = 1 -9X1 + 2x2 + 5X3 = 5 4x1 + X2 - 6X3 = 9 4x1 + X2 - 6X3 = 45 It can be shown that the...

Let v1, . . . , vn be vectors in a vector space V and let T: V W be a linear transformation. (a) If {T(v1) , . . . , T(vn)} is linearly independent in W, show that {v1, . . . , vn} is linearly...

7. Let T: V W be a linear transformation, and let V' be a subspace of V. The restriction of T to V' is the function Ty: VW defined by Tv(v) =T(v) for all v V'. Prove that the restriction Tv is a...

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...

If an economy has 10 million people who are unemployed, the unemployment rate is 20%, and there are 70 million working-age people in the economy as a whole, then what is the number of people not in...

At the instant shown the tensions in the vertical ropes AB and DE are 300N and 200N, respectively. Knowing that the mass of the uniform bar BE is 5kg, determine, at this instant, (a) The force P, (b)...

A 1 % decline in yield will have the least effect on the price of a bond with a _ _ _ _ _ _ _ _ _ _ blank. Multiple Choice 1 0 - year maturity, selling at 8 0 2 0 - year maturity, selling at 8 0 2 0...

Name two types of bonds and the purpose of each. (34.2)

The points (7, 13) and (-3, -11) are at the ends of a diameter.

In a recent week, the television networks CBS, ABC, and NBC together averaged a total of 29.1 million viewers. CBS had 1.7 million more viewers than ABC, and NBC had 1.7 million fewer viewers than...

Nielsen Media Research surveys TV-watching habits and provides a list of the 20 most-watched TV programs each week. Each rating point in the survey represents 1,102,000 households. One week "60...

can you help step-by-step so I can enter into Excel are not supported in Tears and have been disabled A1 B E G H K L M N 1 2 3 D 21. Rate of Return. A bond is issued with a coupon of 4% paid...

Assets, Inc., plans to issue $8 million of bonds with a coupon rate of 7 percent, a par value of $1,000, semiannual coupons, and 20 years to maturity. The current market interest rate on these bonds...

Nascent, Inc., acquires 60 percent of Sea-Breeze Corporation for $414,000 cash on January 1, 2012. The remaining 40 percent of the Sea-Breeze shares traded near a total value of $276,000 both before...