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Alexxandr [17]
3 years ago
11

How do I figure this out?

Mathematics
1 answer:
IRINA_888 [86]3 years ago
6 0

Answer:

what do you mean??

Step-by-step explanation:

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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?
aliina [53]

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)

8 0
3 years ago
The cost of an iPhone is $899.99 plus 8.5% tax. What is the cost of tax for the iPhone?
Sliva [168]

Answer:

724.0915

Step-by-step explanation:

multiply 899.99 and 8.5 you get 724.0915

hope this helps

7 0
3 years ago
joyce traveled at an average speed of 60 miles per hour for 4.5 hours and then traveled at an average speed of 35 miles per hour
SVEN [57.7K]

Answer:

322 1/2 miles

Step-by-step explanation:

Recall how s = d/t ; speed = distance/time.

To effectively solve for distance traveled, rearrange the formula so that distance is on the left-hand side by multiplying speed, and time.

speed × time = distance/time × time

speed × time = distance

distance = speed × time ; d = st.

Now to find the total distance traveled, individually multiply the rate of speed with the time. Time will cancel, and we will get the total mileage.

Given an average speed of 60 mph at a certain time for 4.5 hours, and another average speed of 35 mph at another time of 1.5. Calculate the distance individually, and then add both distances together. Distance 1 = 60 mph × 4.5 h = 60 miles / h × 4.5h = 60 miles × 4.5 = 270 miles.

Distance 2 = 35mph × 1.5h = 35 miles / h × 1.5h = 35 miles × 1.5 = 52 1/2 miles.

Distance 1 + Distance 2 = Total distance.

270 miles + 52 1/2 miles = 322 1/2 miles.

8 0
2 years ago
Preform the indicated operation 15 - (-11)
xenn [34]

Answer:It would be: 15 - (-11)

= 15 + 11

= 26

Step-by-step explanation:

5 0
3 years ago
A sphere is inscribed in a cylinder. The surface area of the sphere and the lateral area of the cylinder are equal. true or fals
Volgvan
It is true that the surface area of the sphere and the lateral area of the cylinder are equal. The answer is True. 
8 0
3 years ago
Read 2 more answers
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