Question: Let U, V be two vector spaces over the same field F. Let W be a subspace of U. We define W' L(U, V )

Let U, V be two vector spaces over the same field F. Let W be a subspace of U. We define W' L(U, V ) by W' = {T L(U, V ) | w W, T(w) = 0}.

a. Show that W' is a subspace of L(U, V ).

b. If both U and V are finite dimensions express dim(W') in terms of dim(U), dim (V), and dim (W). Give proof of your formula. Hint: Consider a matrix representation of T W'.

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

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, V, W be vector spaces over the same field, and let f, g be linear maps as follows: Uv4 w, with the property that gof = 0. 1. Prove that Im(f) C Null(g), and give an explicit example of vector...

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....

Question 4 (15 p) Assume that national income, Y, is exogenously set at one fixed level (full employment) in both the short-run and in the long-run, and that the exchange rate is floating. Assume...

2. (a) Let V be the subspace of Mat22 (R) spanned by 1 0 -20 -4 {(+49) (4) (13) (439)}. 2 -1 2 (i) Find a basis for V consisting of matrices from those listed above. What is the dimension of V? (ii)...

HHelp me to solve the questions as attached below. 3. Consider an economy with A consumers and B rms. Each consumer is endowed with one unit of time and one piece of capital. A representative...

(Requires the subsection on Combining Subspaces, which is optional.) Let U and W be vector spaces. Define a new vector space, consisting of the set U Ã W = {(, ) | U and W} along with these...

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...

Let V be a vector space of dimension n over F. Recall that V V denotes the vector space (also over F) of 2-tensors, i.e. formal linear combinations of bilinear expressions of the form u v for u, v ...

Q4 (10 points) Let V be a vector space of dimension n over F. Recall that V V denotes the vector space (also over F) of 2-tensors, i.e. formal linear combinations of bilinear expressions of the form...

This question involves the use of AGGREGATE linear PYTHOIN regression on the Auto data set. (a) Perform a simple linear regression with mpg as the response and horsepower as the predictor. Describe...

f(n)= n? / log n and g(n) = n(log n)? Select one: O a. f=(g) O b. f=N(g) O c. f=O(g) d. none of these

The management of Advanced Alternative Power Inc. is considering two capital investment projects. The estimated net cash flows from each project are as follows: The wind turbines require an...

In which circumstance would a manager be subject to personal independence restrictions with respect to Company X ? 1 . The manager provides 8 hours of consulting services to Company X . 2 . The...

Using Excel, plot the yield locus using Hills anisotropic yield function for R=0.8, 1.0, 1.5, 2.0, 4, and 6. Use eqn (3-31) for plane stress and a yield stress of 100 MPa in the 2 direction (Y). What...