More from my site

• Every Plane Through the Origin in the Three Dimensional Space is a Subspace Prove that every plane in the $3$-dimensional space $\R^3$ that passes through the origin is a subspace of $\R^3$.   Proof. Each plane $P$ in $\R^3$ through the origin is given by the equation \[ax+by+cz=0\] for some real numbers $a, b, c$. That is, the […]
• Every Group of Order 72 is Not a Simple Group Prove that every finite group of order $72$ is not a simple group. Definition. A group $G$ is said to be simple if the only normal subgroups of $G$ are the trivial group $\{e\}$ or $G$ itself. Hint. Let $G$ be a group of order $72$. Use the Sylow's theorem and determine […]
• If the Kernel of a Matrix $A$ is Trivial, then $A^T A$ is Invertible Let $A$ be an $m \times n$ real matrix. Then the kernel of $A$ is defined as $\ker(A)=\{ x\in \R^n \mid Ax=0 \}$. The kernel is also called the null space of $A$. Suppose that $A$ is an $m \times n$ real matrix such that $\ker(A)=0$. Prove that $A^{\trans}A$ is […]
• Find an Orthonormal Basis of $\R^3$ Containing a Given Vector Let $\mathbf{v}_1=\begin{bmatrix} 2/3 \\ 2/3 \\ 1/3 \end{bmatrix}$ be a vector in $\R^3$. Find an orthonormal basis for $\R^3$ containing the vector $\mathbf{v}_1$.   The first solution uses the Gram-Schumidt orthogonalization process. On the other hand, the second […]
• Idempotent Matrices are Diagonalizable Let $A$ be an $n\times n$ idempotent matrix, that is, $A^2=A$. Then prove that $A$ is diagonalizable.   We give three proofs of this problem. The first one proves that $\R^n$ is a direct sum of eigenspaces of $A$, hence $A$ is diagonalizable. The second proof proves […]
• Symmetric Matrices and the Product of Two Matrices Let $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings. (a) The product $AB$ is symmetric if and only if $AB=BA$. (b) If the product $AB$ is a diagonal matrix, then $AB=BA$.   Hint. A matrix $A$ is called symmetric if $A=A^{\trans}$. In […]
• True or False. The Intersection of Bases is a Basis of the Intersection of Subspaces Determine whether the following is true or false. If it is true, then give a proof. If it is false, then give a counterexample. Let $W_1$ and $W_2$ be subspaces of the vector space $\R^n$. If $B_1$ and $B_2$ are bases for $W_1$ and $W_2$, respectively, then $B_1\cap B_2$ is a […]
• Probability that Alice Tossed a Coin Three Times If Alice and Bob Tossed Totally 7 Times Alice tossed a fair coin until a head occurred. Then Bob tossed the coin until a head occurred. Suppose that the total number of tosses for Alice and Bob was $7$. Assuming that each toss is independent of each other, what is the probability that Alice tossed the coin exactly three […]