Solve the equation:$|x-1|=x-1$ My solution: Case 1 :$ x\ge1$, Hence $x-1=x-1$, therefore infinite solution Case 2 :$ x<1$, Hence $1-x=x-1$,$x=1$, hence...

I'm reading some lecture notes on homotopy, and the author has just proved the theorem: If $f_{0} \simeq f_{1}$ and $g_{0} \simeq g_{1}$, then $g_{0}...

I want to calculate the skewness of a Poisson distributed random variable but I can't find my mistake. $$\begin{align} \mathbb{E}(X^3)&= (-1)^3 \...

Let $A$ be a $4 \times 7$ real matrix and $B$ be a $7 \times 4$ real matrix such that $AB=I_4$. Which of the following are true? 1) rank$(A)=4$ 2) rank$(B...

What is the importance of eigenvalues/eigenvectors?

I was asked to state that the claim is true or false, I must give a prove to say it is true and counter example if it is false. However I say it is True;T...

I am working through Cohomology of Groups (Kenneth Brown) and I noticed that in the beginning chapters when tensor products came up, he would emphasize th...

This page says- absolute change = new quantity – old quantity. For ex- Example 1: On September 20, a gallon of gas at my usual gas station cost $1.83. On ...

$\mathbf{Theorem}$. Let $A \subset \mathbb{R}$ be a measurable set, $\mu(A)<\infty$ and $f$ a measurable function with domain $A$. Then, for every $\va...

If I have the equation $9x^2-4y^2-72x=0 $ and I know that is a hyperbola, how would I find the standard form for this equation? I'm not sure how to c...