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andrew-mc [135]
3 years ago
14

You spend 30 minutes reading emails. You then spend 3 hours watching television. write the ratio of the amount of time spent rea

ding emails to the amount of time spent watching television as a fraction in simplest form.
Mathematics
1 answer:
oksano4ka [1.4K]3 years ago
8 0

Ten minutes to 1 hour

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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
anne picks a 4-digit number the first number is not 0 the four digit number is a multiple of 5 how many different 4-digit number
Gala2k [10]
The first digit of a number can be 1-9. The 2nd and the 3rd digit can be 1-10. The last digit can be 0 or 5 ( 2 numbers ):
9 * 10 * 10 * 2 = 1,800
She could pick 1,800 different 4-digit numbers.
8 0
3 years ago
What is the difference between the mean and the median of the data set?<br> (22,8 10,18,12, 20)
Lera25 [3.4K]

Answer:

mean is 8 and median is 15

Step-by-step explanation:

6 0
3 years ago
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What times what equals 225?
defon
I got 5 times 45 because I multiplied by 5 to get my answer.
4 0
3 years ago
Two numbers are in the ratio 3:7
lbvjy [14]

Answer:

98 And 18 i think

Step-by-step explanation:

8 0
3 years ago
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