Explanation Given information: The transformation T is an isomorphism from V to W . Concept used: A function T from V to W is said to be linear transformation if, T ( f + g ) = T ( f ) + T ( g ) and T ( k f ) = k T ( f ) for all elements f and g of V and for all scalars k . Proof: Since, the transformation T is an isomorphism from V to W , so it is injective and surjective. If T is injective, then for all y 1 , y 2 ∈ V where y 1 ≠ y 2 , then T ( y 1 ) ≠ T ( y 2 ) . This implies that there exists T − 1 . For proving T − 1 is isomorphism, first is required to show that it is a linear transformation. Suppose that, x 1 , x 2 ∈ W . Since, T is an isomorphism, so there exists fixed y 1 , y 2 ∈ V such that T ( y 1 ) = x 1 and T ( y 2 ) = x 2 . Now, T − 1 ( k 1 x 1 + k 2 x 2 ) = T − 1 ( k 1 T ( y 1 ) + k 2 T ( y 2 ) ) = T − 1 ( T ( k 1 y 1 ) + T (

5th Edition

ISBN: 9780321796943

Author: BRETSCHER, Otto.

Publisher: Pearson Education

See similar textbooks

Videos

Question

Chapter 4.2, Problem 76E

To determine

To prove:if T is an isomorphism from V to W , then T−1 is an isomorphism from W to V .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.2, Problem 75E

Chapter 4.2, Problem 77E

Students have asked these similar questions

Assume that you fancy polynomial splines, while you actually need ƒ(t) = e²/3 – 1 for t€ [−1, 1]. See the figure for a plot of f(t). Your goal is to approximate f(t) with an inter- polating polynomial spline of degree d that is given as sa(t) = • Σk=0 Pd,k bd,k(t) so that sd(tk) = = Pd,k for tk = −1 + 2 (given d > 0) with basis functions bd,k(t) = Σi±0 Cd,k,i = • The special case of d 0 is trivial: the only basis function b0,0 (t) is constant 1 and so(t) is thus constant po,0 for all t = [−1, 1]. ...9 The d+1 basis functions bd,k (t) form a ba- sis Bd {ba,o(t), ba,1(t), bd,d(t)} of the function space of all possible sα (t) functions. Clearly, you wish to find out, which of them given a particular maximal degree d is the best-possible approximation of f(t) in the least- squares sense. _ 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 function f(t) = exp((2t)/3) - 1 to project -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5…

An image processor considered a 750×750 pixels large subset of an image and converted it into gray-scale, resulting in matrix gIn - a false-color visualization of gIn is shown in the top-left below. He prepared a two-dim. box filter f1 as a 25×25 matrix with only the 5×5 values in the middle being non-zero – this filter is shown in the top-middle position below. He then convolved £1 with itself to get £2, before convolving £2 with itself to get f3. In both of the steps, he maintained the 25×25 size. Next, he convolved gIn with £3 to get gl. Which of the six panels below shows g1? Argue by explaining all the steps, so far: What did the image processor do when preparing ₤3? What image processing operation (from gin to g1) did he prepare and what's the effect that can be seen? Next, he convolved the rows of f3 with filter 1/2 (-1, 8, 0, -8, 1) to get f4 - you find a visualization of filter f 4 below. He then convolved gIn with f4 to get g2 and you can find the result shown below. What…

3ur Colors are enchanting and elusive. A multitude of color systems has been proposed over a three-digits number of years - maybe more than the number of purposes that they serve... - Everyone knows the additive RGB color system – we usually serve light-emitting IT components like monitors with colors in that system. Here, we use c = (r, g, b) RGB with r, g, bЄ [0,1] to describe a color c. = T For printing, however, we usually use the subtractive CMY color system. The same color c becomes c = (c, m, y) CMY (1-c, 1-m, 1-y) RGB Note how we use subscripts to indicate with coordinate system the coordinates correspond to. Explain, why it is not possible to find a linear transformation between RGB and CMY coordinates. Farbenlehr c von Goethe Erster Band. Roſt einen Defte mit fergen up Tübingen, is et 3. Cotta'fden Babarblung. ISIO Homogeneous coordinates give us a work-around: If we specify colors in 4D, instead, with the 4th coordinate being the homogeneous coordinate h so that every actual…

Chapter 4 Solutions

Linear Algebra With Applications

Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

To prove: if T is an isomorphism from V to W , then T − 1 is an isomorphism from W to V .

Solution Summary: The author explains that the transformation T is an isomorphism from V to W, so it is injective and surjective.

Videos

To prove:if T is an isomorphism from V to W , then T−1 is an isomorphism from W to V .

Want to see the full answer?

Chapter 4 Solutions