Question: Let U and V be vector spaces over a field F, and assume that U is non-trivial (i.e. has at least one non-zero vector)

Let U and V be vector spaces over a field F, and assume that U is non-trivial (i.e. has at least one non-zero vector) and finite- dimensional. Let {,...,} be a linearly independent set in U, and let V EV. Prove that there exists a linear transformation f: U V such that f(u) = V f(uk) Vk. V1,...,

Step by Step Solution

★★★★★

3.52 Rating (145 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

ANSWER Since uu is linearly independent and U is finitedimensional we can extend uu to a basis uu vv ... 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

Students Have Also Explored These Related Mathematics Questions!

Let U and V be vector spaces over the same field F. We will create a new, larger, vector space, which contains both of U and V as subspaces. This new vector space is called the direct sum of U and V....

Prove or disprove the following statements. (a) Let V and W be finite dimensional vector spaces over a field K such that dim V dim W. Then any linear map T: V W has a nonzero kernel. (b) Any subset...

1 Let V and W be vector spaces over F with V finite-dimensional, and let U be any subspace of V. Given a linear map SE L(U,W), prove that there exists a linear map TE L(V, W) such that, for every u e...

In Problems 5994, solve each inequality. Express your answer using set notation or interval notation. Graph the solution set. 2x2 3 + x

SoftWear, Limited (SWL), is a continuing case study that illustrates the knowledge and skills described in each chapter. In this case study, the student acts as a member of the SWL systems...

The following table lists the average monthly cost to workers for family health insurance for various years. a) Model the data with a linear function. Let the independent variable represent the...

In the examination of a Probability I course at aUniversity, 180 students participated in the exam. Among these students, 80 study for a Mathematics degree, 60 for a Statistics degree, and 40 are in...

Consider the data in the Excel file Nuclear Power. Use simple linear regression to forecast the data. What would be the forecasts for the next 3 years?

Feather Friends, Inc., distributes a high-quality wooden birdhouse that sells for $40 per unit. Variable expenses are $20.00 per unit, and fixed expenses total $180,000 per year. Its operating...

Sam Mc Kenzy is a founder and CEO of McKenzy Restaurants Inc., a regional company. Sam is considering opening several new restaurants. Sally Thorton, the company CFO has been put in charge of the...

Need steps on how to solve 7. Secant Lines Consider the function f(x) = Vx and the point P(4, 2) on the graph of f. (a) Graph f and the secant lines passing through P(4, 2) and O(x, f (x)) for...

we look at how companies use CRM, SCM, and ERP to gain a competitive advantage and run more efficiently. Find an industry/business or company currently using one of those technologies (CRM, SCM,...

1. What are the practices that Xerox used in its TQM implementation? Discuss. 2. What are some barriers that you think Xerox has faced during TQM implementation? Relate them. 3. Does TQM...

An automobile manufacturer based in Lexington, Kentucky, wants to move production jobs from Japan to Mexico so that it can be more responsive to changing demands in its North American markets. Which...

Two weeks after they were told about the city's proposal to demolish the neighborhood playground and build a parking lot, Diana approached Tony as they were heading to another neighborhood...

#3. Test to see if there is a significant difference between the age of the participant and the age of the partner. Use a paired-sample t-test and an alpha level of 1%. How would you interpret the...

Royal Lawncare Company produces and sells two packaged products-Weedban and Greengrow. Revenue and cost information relating to the products follow: Weedban Greengrow $12.00 35.00 $2.90 12.00 selling...

Answer the following two independent questions. a. MM Corporation is considering several proposed investments for the coming budget year. MM produces electrical apparatus for industrial complexes....

Find all the eigenvalues and associated eigenvectors of each of the following matrices: (a) (b) (c) (d) 001 1i0 101 010 900 0-0 001

Let L: V V be a linear operator defined by L(v) = cv, where c is a fixed constant. Prove that the representation of L with respect to any ordered basis for V is a scalar matrix.

Use the Gram-Schmidt process to find an orthonormal basis for the subspace of R4 with basis 0101 1100 1001

1. Conduct a simple replication of the better-thanaverage effect by surveying your classmates about how they rate themselves with regard to others on some characteristic. If possible do the same...

3. Are indigenous personalities really indigenous? For example, the Japanese culture includes the concept of amae described above. Do you think that amae also exists in other cultures, at least in...

1. Consider a hypothetical research study that you might like to conduct that would investigate gender differences of a particular behavior or psychological phenomenon. How would you conduct the...