Math Is Fun Forum

We have a linear transformation from a matrix space, to a real space defined as a product of all elements in the matrix.
How can we define a matrix representing such transformation with respect to the standard basis?

The 'standard basis' here is a set of matrices:

But the result is not a matrix.

So I can write something like that:

But how to define a matrix representing such transformation?

Problem:
Let

be a square invertible matrix. Let

be the linear transformation defined by

. Prove that

is an isomorphism.

How to start the proof?

I am thinking about something like this: Let

, then by definition of matrix invertibility

. So

.

But there is an issue of order, it does matter in the matrix product and I do violate it in this proof. So it must be wrong. But I do not see how to fix it.

We have two spheres with centers (x_1, y_1, z_1) and (x_2, y_2, z_2). And radii r_1 and r_2.
What is the volume of solid intersection of these two spheres?

I am not sure how to approach this question.
I know that a point will belong to the solid iff:

But how to find a volume limited by this surface?
It is not a curve, so I cannot use  a straight 'solid of revolution' formulas, can I?

It should be something like a sum of all points which satisfy the inequality:

but how?

We have two  three digit binary numbers. In one number all digits do invert every single step (does not matter what the initial value was, lets say we have a 000-111).
In another number, each digit inverts on random.
Each inversion requires some work to be done. So for the first number, in 10 steps we require 3*10=30 energy.
If none of the digits in the second number was inverted (this can happen), then in the same 10 steps we require 0*10=0 energy. If all of the digits in the second number was inverted (this also can happen), then we will require 3*10=30 energy.
But since the second number is under randomizer we will usually have less energy spent on it in comparison with the first number.
The question is: what is the average reduction of energy which would be required by the second number in comparison with the first one?

My current train of thought is:

There are four possible cases:
0 inverted - 2^3 combinations
1 inverted - (2^3)^3 combinations
2 inverted - (2^3)^3 combinations
3 inverted - 2^3 combinations

At each step one of those four cases can happen.
So the probability to spend get any single from combination A to combination B is

We have

probability to spend 0 energy at one step
We have

probability to spend 1 energy at one step
same for 2 and 3.

So on average we supposed to get

In other words, the second number will spend 1.5 energy to invert random number of bits and on average we will spend 1.5*10=15 energy on the second number.

Is this correct?