Question: Let T: V W be a linear transformation between two finite-dimensional vector spaces. (a) Prove that if dim V < dim W, then T

Let T: V → W be a linear transformation between two finite-dimensional vector spaces.
(a) Prove that if dim V < dim W, then T cannot be onto.
(b) Prove that if dim V > dim W, then T cannot be one-to-one.

Step by Step Solution

★★★★★

3.36 Rating (165 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a Suppose that T V W and that dim V dimW Then rangeT is a su... 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 (2791).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

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 L: V W be a linear transformation from a vector space V into a vector space W. The image of a subspace V1 of V is defined as L(V1) = {w in W | w = L(v) for some v in V}. Show that L(V1) is a...

Let T: V W be a linear transformation. (a) If U is a subspace of V, show that T(U) = (T(u) | u in U} is a subspace of W (called the image of U under T). (b) If P is a subspace of W, show that {v in...

Need help with the question, the question is shown below. 1. Let T : V > W be a. linear map of nite dimensional vector spaces. Prove the following facts. You can use results from the previous problem...

Let L: V W be a linear transformation, and let dim V = dim W. Prove that L is invertible if and only if the image of a basis for V under L is a basis for W.

please submit with handwritten answer. otherwise please don't try . If W is a subspace of an inner product space V, and if {w1, W2, " . ., we} is an orthonormal basis of W, then given any r E...

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...

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...

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...

SOME DUAL SPACE QUESTIONS 1. Let T : V ! W be an onto map of finite dimensional vector spaces. Prove that T : V ! W is one to one. 2. Let F be a field and T : M22 (F ) ! F be the trace map. For any...

How is net income determined?

Culver Inc. declared a cash dividend of $0.50 per share on its 1.60 million outstanding shares. The dividend was declared on August 1 and is payable on September 9 to all shareholders of record on...

Which of the following answer choices best describes income in respect of a decedent? a. Income earned or received by the decedent up to 30 days after the date of death b. Income pledged by others to...

What kinds of fi nancial information are presented on an income statement, a balance sheet, and a cash fl ow statement? (36.2)

Sometimes the graph of a quadratic equation is a straight line, a pair of straight lines, or a single point. We refer to such a graph as a degenerate conic. It is also possible that the equation is...

Identify the quadric with the given equation and give its equation in standard form. 2xy + z = 0

Identify the quadric with the given equation and give its equation in standard form. 16x 2 + 100y 2 + 9z 2 - 24xz - 60x - 80z = 0

Which of the following statements is True? Select one: The floating rate regularly adjusts to a reference rate. O b. None of the options are correct. O c. The term 'spot loan' refers to a loan that...

Please answer question 2 in excel, with cell reference please! If a firm has a payout ratio of 6 5 % and a ROE on projects of 1 1 % , what is the growth rate for the firm? Assume a firm has a cost of...

HELP PLEASE (Individual or component costs of capital) Compute the cost of capital for the firn for the folowing a. Currently bonds with a simlar credt rating and matunty as the firris cutatanding...