You would divide 1/16 by 3/8
Yes. Conceptually, all the matrices in the group have the same structure, except for the variable component
. 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]](https://tex.z-dn.net/?f=%20%5Cphi%3A%5C%20%5Cmathbb%7BR%7D%20%5Cto%20G%2C%5Cquad%20%5Cphi%28x%29%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%26x%5C%5C0%261%5Cend%7Barray%7D%5Cright%5D%20)
is an isomorphism.
First of all, it is injective: suppose
. Then, you trivially have
, 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]](https://tex.z-dn.net/?f=%20%5Cphi%28x%29%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%26x%5C%5C0%261%5Cend%7Barray%7D%5Cright%5D%2C%5Cquad%20%5Cphi%28y%29%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%26y%5C%5C0%261%5Cend%7Barray%7D%5Cright%5D%20)
Secondly, it is trivially surjective: the matrix
![\phi(x) = \left[\begin{array}{cc}1&x\\0&1\end{array}\right]](https://tex.z-dn.net/?f=%20%5Cphi%28x%29%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%26x%5C%5C0%261%5Cend%7Barray%7D%5Cright%5D%20)
is clearly the image of the real number x.
Finally,
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)](https://tex.z-dn.net/?f=%20%5Cphi%20%28x%2By%29%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%26x%2By%5C%5C0%261%5Cend%7Barray%7D%5Cright%5D%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%26x%5C%5C0%261%5Cend%7Barray%7D%5Cright%5D%20%5Ccdot%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%26y%5C%5C0%261%5Cend%7Barray%7D%5Cright%5D%20%3D%20%5Cphi%28x%29%20%5Ccdot%20%5Cphi%28y%29)
x = -1/3
Explanation:
-3x-2=-1
collect like terms by adding +2 to both sides:
-3x -2 + 2 = -1 + 2
-3x = 1
Divide both sides by the coefficient of x:
coefficient of x = -3
-3x/-3 = 1/-3
x = -1/3
Answer:
Surface Area: 310 square inches.
Step-by-step explanation:
There are two ways to do this:
A) Formula for Surface Area of a Rectangular Prism = 2 * ( l*w + w*h + h*l). Where l is length, w is width & h is height. Based on the question:
l = 10 inch
w = 5 inch
h = 7 inch
Surface Area = 2 * ( 10*5 + 5*7 + 7*10) = 310 square inches.
B) Formula for Area of Rectangle = l*w, where l is length & w is width.
Look at the picture, I have marked the corners O,P,Q,R,S,T,U,V,W,X,Y,Z
If we calculate the Area of each rectangle and add them all we will get the surface area automatically.
- Area of PQRS = 10*7 = 70 square inches
- Area of STUV = 7*5 = 35 square inches
- Area of VWXY = (7+5)*10 = 120 square inches
- Area of ORYZ = 7*5 = 35 square inches
- Area of RSVY = 10*5 = 50 square inches
Now add them all = 70+35+120+35+50 = 310 square inches.
Answer:
Yes; $23.10
Step-by-step explanation:
5% of 22 is 1.10 and if you add the tax to the give price it is less than $24