Recall that a relation is an equivalence relation if and only if is symmetric, reflexive and transitive. In order to simplify the notation we will use A↔B when A is in relation with B.

Reflexive: We need to prove that A↔A. Let us write J for the identity matrix and recall that J is invertible. Notice that $A=J^{-1}AJ$ . Thus, A↔A.

Symmetric: We need to prove that A↔B implies B↔A. As A↔B there exists an invertible matrix P such that $A=P^{-1}BP$ . In this equality we can perform a right multiplication by $P^{-1}$ and obtain $AP^{-1} =P^{-1}B$ . Then, in the obtained equality we perform a left multiplication by P and get $PAP^{-1} =B$ . If we write $Q=P^{-1}$ and $Q^{-1} = P$ we have $B = Q^{-1}AQ$ . Thus, B↔A.

Transitive: We need to prove that A↔B and B↔C implies A↔C. From the fact A↔B we have $A=P^{-1}BP$ and from B↔C we have $B=Q^{-1}CQ$ . Now, if we substitute the last equality into the first one we get

$A=P^{-1}Q^{-1}CQP = (P^{-1}Q^{-1})C(QP)$ .

Recall that if P and Q are invertible, then QP is invertible and $(QP)^{-1}=P^{-1}Q^{-1}$ . So, if we denote R=QP we obtained that

$A=R^{-1}CR$ . Hence, A↔C.

Therefore, the relation is an equivalence relation.

4 0

3 years ago

Find the volume of the sphere with a radius, r, of 3/2 feet and π ≈ 3.14.

nalin [4]

Answer:

V= 14.13

Step-by-step explanation:

Information needed:

V= 4/3 $\pi$ r^3

$\pi$ = 3.14

r= 3/2

Solve:

V= 4/3 $\pi$ r^3

V= 4/3(3.14)(3/2)^3

V= 4/3(3.14)(27/8)

V= 14.13

3 0

3 years ago

Read 2 more answers

Plz help me I had to put it was math Bc there is no science option plz help me thank you.

SVEN [57.7K]

Answer:

see the attached

Step-by-step explanation:

To fill in the data table, you need information about the layers of the Earth. That information is provided in the second attachment.

Among other things, it shows the density of the rock layers varies over a range of almost 6 to 1 from the top layer to the inner core. Your representative liquids vary in density over a range of about 1.8 to 1, so clearly cannot match the relative densities.

In the attached spreadsheet, we have used the average of the top and bottom density values for each layer to represent the density of the layer. Then we did a linear mapping so that the minimum density maps to lamp oil, and the maximum density maps to honey. (You get slightly different results if you use a different mapping strategy.)

For your convenience in stacking your "density cylinder", we have shown the relative thickness of each of the layers as a percentage of the overall radius of the Earth. That is, if your cylinder is 100 mL deep, for example, then the top layer will be 0.5 mL deep, the next layer 11.3 mL deep, and so on.

We have elected not to use alcohol for the top layer, because it is miscible with all of the liquids immediately below that in the table. That is, you could not maintain alcohol as a separate layer in your cylinder because it would dissolve into the layer below. It might take a little effort (or care) to keep the lamp oil separated from the vegetable oil, as we expect those might try to mix, as well.

If you compare the scaled density values with the values for the "representative" liquids we have chosen, you will see that some are closer than others. The idea is to try to get the best match, while also choosing liquids that don't mix well.

7 0

3 years ago

which properties are present in a table that represents an exponential function in the form y=b^x when b>1

vladimir1956 [14]

Answer:

The function has the property of exponential growth function.

Step-by-step explanation:

The exponential function is given by $y = b^{x}$ , where b > 1.

Now, for x = 0, we get y = 1,

For x = 1, we get y = b

For, x = 2, we get y = b²

And so on.

Now, as b > 1, so, b² > b > 1 and the values of y are increasing as the value of x increases.

Therefore, the function has the property of exponential growth function.

3 0

4 years ago