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• Find a Basis for the Subspace spanned by Five Vectors Let $S=\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\}$ where \[ \mathbf{v}_{1}= \begin{bmatrix} 1 \\ 2 \\ 2 \\ -1 \end{bmatrix} ,\;\mathbf{v}_{2}= \begin{bmatrix} 1 \\ 3 \\ 1 \\ 1 \end{bmatrix} ,\;\mathbf{v}_{3}= \begin{bmatrix} 1 \\ 5 \\ -1 […]
• Diagonalizable Matrix with Eigenvalue 1, -1 Suppose that $A$ is a diagonalizable $n\times n$ matrix and has only $1$ and $-1$ as eigenvalues. Show that $A^2=I_n$, where $I_n$ is the $n\times n$ identity matrix. (Stanford University Linear Algebra Exam) See below for a generalized problem. Hint. Diagonalize the […]
• A One Side Inverse Matrix is the Inverse Matrix: If $AB=I$, then $BA=I$ An $n\times n$ matrix $A$ is said to be invertible if there exists an $n\times n$ matrix $B$ such that $AB=I$, and $BA=I$, where $I$ is the $n\times n$ identity matrix. If such a matrix $B$ exists, then it is known to be unique and called the inverse matrix of $A$, denoted […]
• If a Matrix $A$ is Singular, There Exists Nonzero $B$ such that the Product $AB$ is the Zero Matrix Let $A$ be an $n\times n$ singular matrix. Then prove that there exists a nonzero $n\times n$ matrix $B$ such that \[AB=O,\] where $O$ is the $n\times n$ zero matrix.   Definition. Recall that an $n \times n$ matrix $A$ is called singular if the […]
• Use Lagrange’s Theorem to Prove Fermat’s Little Theorem Use Lagrange's Theorem in the multiplicative group $(\Zmod{p})^{\times}$ to prove Fermat's Little Theorem: if $p$ is a prime number then $a^p \equiv a \pmod p$ for all $a \in \Z$.   Before the proof, let us recall Lagrange's Theorem. Lagrange's Theorem If $G$ is a […]
• Group Homomorphism Sends the Inverse Element to the Inverse Element Let $G, G'$ be groups. Let $\phi:G\to G'$ be a group homomorphism. Then prove that for any element $g\in G$, we have \[\phi(g^{-1})=\phi(g)^{-1}.\]     Definition (Group homomorphism). A map $\phi:G\to G'$ is called a group homomorphism […]
• 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 […]
• A Ring is Commutative if Whenever $ab=ca$, then $b=c$ Let $R$ be a ring and assume that whenever $ab=ca$ for some elements $a, b, c\in R$, we have $b=c$. Then prove that $R$ is a commutative ring.   Proof. Let $x, y$ be arbitrary elements in $R$. We want to show that $xy=yx$. Consider the […]