1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
AnnyKZ [126]
3 years ago
13

Let g' be the group of real matricies of the form [1 x 0 1]. Is the map that sends x to this matrix an isomorphism?

Mathematics
1 answer:
aliina [53]3 years ago
8 0

Yes. Conceptually, all the matrices in the group have the same structure, except for the variable component x. So, each matrix is identified by its top-right coefficient, since the other three entries remain constant.

However, let's prove in a more formal way that

\phi:\ \mathbb{R} \to G,\quad \phi(x) = \left[\begin{array}{cc}1&x\\0&1\end{array}\right]

is an isomorphism.

First of all, it is injective: suppose x \neq y. Then, you trivially have \phi(x) \neq \phi(y), because they are two different matrices:

\phi(x) = \left[\begin{array}{cc}1&x\\0&1\end{array}\right],\quad \phi(y) = \left[\begin{array}{cc}1&y\\0&1\end{array}\right]

Secondly, it is trivially surjective: the matrix

\phi(x) = \left[\begin{array}{cc}1&x\\0&1\end{array}\right]

is clearly the image of the real number x.

Finally, \phi and its inverse are both homomorphisms: if we consider the usual product between matrices to be the operation for the group G and the real numbers to be an additive group, we have

\phi (x+y) = \left[\begin{array}{cc}1&x+y\\0&1\end{array}\right] = \left[\begin{array}{cc}1&x\\0&1\end{array}\right] \cdot \left[\begin{array}{cc}1&y\\0&1\end{array}\right] = \phi(x) \cdot \phi(y)

You might be interested in
Help me plsss its pretty easy lol
DaniilM [7]

Answer:

D, or the last one.

Step-by-step explanation:

Brainliest?

4 0
3 years ago
Read 2 more answers
A scale model of a boat is 6 inches long. The actual boat is 42 feet long. Which statement(s) are true??
Vitek1552 [10]

Answer:

B. 1 in = 7 ft

Step-by-step explanation:

Since on the scale the boat was 6 inches and the actual boat was 42 feet, we need to divide to see how many feet correlate to the number on inches on the scale. So we do 42 / 6, which equals 7. This means that for every inch on the scale, it will be 7 feet in real life.

3 0
3 years ago
Read 2 more answers
A postage stamp is shaped like a rectangle with a perimeter of 88 millimeters. The length (x) is 10 millimeters less than twice
Oxana [17]
The answer would be D. It says the length is 10 millimeters less than TWICE the width, so it would be 10 millimeters greater than the original width.
5 0
3 years ago
A navigator plots the course a plane is currently traveling. The plane is 300 miles from its destination. If it continues on its
inna [77]
We have to use the rule of cosx° to solve this problem. Attached is a diagram of the navigator's course for the plane. It is similar to the shape of a triangle. We know the plane is 300 miles from its destination, so that will be one of the sides. On the current course, it is 325 miles from its destination, so that will be another one of the sides. The last side is 125 because that is the distance between the destination and the anticipated arrival. Cosx° is what we are looking for. 

To find how many degrees off course the plane is, we must use the rules of Cosx°, which is shown in the attached image. 

The plane is approximately 23° off course. 

3 0
3 years ago
The expression x + 18 represents the perimeter of the rectangle shown. Write an expression that represents the length of the rec
IgorLugansk [536]

Answer:

can you add a picture

Step-by-step explanation:

5 0
2 years ago
Other questions:
  • U xy, for u = 2, x = 9, and y = 6
    15·1 answer
  • Use the zero product property to find the solutions to the equation
    14·2 answers
  • Solve for the indicated variable 3x - y = 10; y
    6·1 answer
  • Please help me with these. These are so hard.<br><br>​
    11·1 answer
  • Find the lateral area for the pyramid with the equilateral base.<br><br>144<br>180<br>288
    8·2 answers
  • Andrea has a savings account with $80 that earns 2% interest per
    5·1 answer
  • X=8 y=4 z=0 w=2 <br><br> W(x-2z)=
    13·1 answer
  • What's 20000 times a million thousand
    10·1 answer
  • Find the measure of the indicated angle to the nearest degree.
    6·2 answers
  • Solve each problem and decide which is the best of the choices given.
    11·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!