# Top 10 Popular Math Problems in 2016-2017

It’s been a year since I started this math blog!!

More than 500 problems were posted during a year (July 19th 2016-July 19th 2017).

I made a list of the 10 math problems on this blog that have the most views.

Can you solve all of them?

The level of difficulty among the top 10 problems.
【★★★】 Difficult (Final Exam Level)
【★★☆】 Standard(Midterm Exam Level)
【★☆☆】 Easy (Homework Level)

## 10th Place: Express a Vector as a Linear Combination of Other Vectors

Problem.【★☆☆】
Express the vector $\mathbf{b}=\begin{bmatrix} 2 \\ 13 \\ 6 \end{bmatrix}$ as a linear combination of the vectors
$\mathbf{v}_1=\begin{bmatrix} 1 \\ 5 \\ -1 \end{bmatrix}, \mathbf{v}_2= \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \mathbf{v}_3= \begin{bmatrix} 1 \\ 4 \\ 3 \end{bmatrix}.$

This is a Linear Algebra exam problem at the Ohio State University.

The solution is given in the post
Express a Vector as a Linear Combination of Other Vectors

## 9th Place: Is an Eigenvector of a Matrix an Eigenvector of its Inverse?

Problem.【★★☆】
Suppose that $A$ is an $n \times n$ matrix with eigenvalue $\lambda$ and corresponding eigenvector $\mathbf{v}$.

(a) If $A$ is invertible, is $\mathbf{v}$ an eigenvector of $A^{-1}$? If so, what is the corresponding eigenvalue? If not, explain why not.

(b) Is $3\mathbf{v}$ an eigenvector of $A$? If so, what is the corresponding eigenvalue? If not, explain why not.

This is a Linear Algebra exam problem at Stanford University.

The solution is given in the post
Is an Eigenvector of a Matrix an Eigenvector of its Inverse?

## 8th Place: The Union of Two Subspaces is Not a Subspace in a Vector Space

Problem.【★★☆】
Let $U$ and $V$ be subspaces of the vector space $\R^n$.
If neither $U$ nor $V$ is a subset of the other, then prove that the union $U \cup V$ is not a subspace of $\R^n$.

The solution is given in the post
The Union of Two Subspaces is Not a Subspace in a Vector Space

## 7th Place: Vector Form for the General Solution of a System of Linear Equations

Problem.【★☆☆】
Solve the following system of linear equations by transforming its augmented matrix to reduced echelon form (Gauss-Jordan elimination). Find the vector form for the general solution.
\begin{align*}
x_1-x_3-3x_5&=1\\
3x_1+x_2-x_3+x_4-9x_5&=3\\
x_1-x_3+x_4-2x_5&=1.
\end{align*}

The solution is given in the post
Vector Form for the General Solution of a System of Linear Equations

## 6th Place: Eigenvalues of a Hermitian Matrix are Real Numbers

Problem.【★★★】
Show that eigenvalues of a Hermitian matrix $A$ are real numbers.

This is again a Linear Algebra exam problem from the Ohio State University.

The proof is given in the post
Eigenvalues of a Hermitian Matrix are Real Numbers

## 5th Place: Find all Values of x such that the Given Matrix is Invertible

Problem.【★★☆】
Let
$A=\begin{bmatrix} 2 & 0 & 10 \\ 0 &7+x &-3 \\ 0 & 4 & x \end{bmatrix}.$ Find all values of $x$ such that $A$ is invertible.

This is again one of the Stanford University Linear Algebra exam problems.

The solution is given in the post
Find all Values of x such that the Given Matrix is Invertible

## 4th Place: How to Diagonalize a Matrix. Step by Step Explanation

Problem.【★★☆】
Diagonalize the matrix
$A=\begin{bmatrix} 4 & -3 & -3 \\ 3 &-2 &-3 \\ -1 & 1 & 2 \end{bmatrix}$ by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

This post explains the procedure of diagonalization of a matrix.
Several exercise problems of diagonalization are given too.

The solution is given in the post
How to Diagonalize a Matrix. Step by Step Explanation

## 3rd Place: Eigenvalues of Real Skew-Symmetric Matrix are Zero or Purely Imaginary and the Rank is Even

Problem.【★★★】
Let $A$ be a real skew-symmetric matrix, that is, $A^{\trans}=-A$.
Then prove the following statements.

(a) Each eigenvalue of the real skew-symmetric matrix $A$ is either $0$ or a purely imaginary number.

(b) The rank of $A$ is even.

The proof is given in the post
Eigenvalues of Real Skew-Symmetric Matrix are Zero or Purely Imaginary and the Rank is Even

## 2nd Place: True or False: $(A-B)(A+B)=A^2-B^2$ for Matrices $A$ and $B$

Problem.【★☆☆】
Let $A$ and $B$ be $2\times 2$ matrices. Prove or find a counterexample for the statement that $(A-B)(A+B)=A^2+B^2$.

The solution is given in the post
True or False: $(A-B)(A+B)=A^2-B^2$ for Matrices $A$ and $B$

## 1st Place: Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to $\R^3$

Problem.【★★☆】
Define the map $T:\R^2 \to \R^3$ by $T \left ( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\right )=\begin{bmatrix} x_1-x_2 \\ x_1+x_2 \\ x_2 \end{bmatrix}$.

(a) Show that $T$ is a linear transformation.

(b) Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for each $\mathbf{x} \in \R^2$.

(c) Describe the null space (kernel) and the range of $T$ and give the rank and the nullity of $T$.

The winner is a problem about linear transformation.
This is one of the most basic problems about linear transformation.

The solution is given in the post
Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to $\R^3$

## Comment

Even though I posted problems in Linear Algebra, Group Theory, Ring Theory, Module Theory, Field Theory, Number Theory, all of the top 10 popular problems are Linear Algebra problems.

After listing the top 10 problems, I realized that the list is just like a linear algebra exam.
Of course 4 of them are actual exam problems from either the Ohio State University or Stanford University.
I also believe that the other problems were used in some exams as they are typical problems in linear algebra.

All 10 problems were posted on or before January 2017. The more recent problems didn’t get enough time to win the top 10.
Let’s see how the ranking will change next year.

If you haven’t, why don’t you solve all the top 10 problems?

Prove that if $2^n-1$ is a Mersenne prime number, then $N=2^{n-1}(2^n-1)$ is a perfect number. On the other hand, prove...