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  • Hyperplane in $n$-Dimensional Space Through Origin is a SubspaceHyperplane in $n$-Dimensional Space Through Origin is a Subspace A hyperplane in $n$-dimensional vector space $\R^n$ is defined to be the set of vectors \[\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}\in \R^n\] satisfying the linear equation of the form \[a_1x_1+a_2x_2+\cdots+a_nx_n=b,\] […]
  • Are Linear Transformations of Derivatives and Integrations Linearly Independent?Are Linear Transformations of Derivatives and Integrations Linearly Independent? Let $W=C^{\infty}(\R)$ be the vector space of all $C^{\infty}$ real-valued functions (smooth function, differentiable for all degrees of differentiation). Let $V$ be the vector space of all linear transformations from $W$ to $W$. The addition and the scalar multiplication of $V$ […]
  • A Recursive Relationship for a Power of a MatrixA Recursive Relationship for a Power of a Matrix Suppose that the $2 \times 2$ matrix $A$ has eigenvalues $4$ and $-2$. For each integer $n \geq 1$, there are real numbers $b_n , c_n$ which satisfy the relation \[ A^{n} = b_n A + c_n I , \] where $I$ is the identity matrix. Find $b_n$ and $c_n$ for $2 \leq n \leq 5$, and […]
  • Every $n$-Dimensional Vector Space is Isomorphic to the Vector Space $\R^n$Every $n$-Dimensional Vector Space is Isomorphic to the Vector Space $\R^n$ Let $V$ be a vector space over the field of real numbers $\R$. Prove that if the dimension of $V$ is $n$, then $V$ is isomorphic to $\R^n$.   Proof. Since $V$ is an $n$-dimensional vector space, it has a basis \[B=\{\mathbf{v}_1, \dots, […]
  • Quiz 12. Find Eigenvalues and their Algebraic and Geometric MultiplicitiesQuiz 12. Find Eigenvalues and their Algebraic and Geometric Multiplicities (a) Let \[A=\begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 &1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{bmatrix}.\] Find the eigenvalues of the matrix $A$. Also give the algebraic multiplicity of each eigenvalue. (b) Let \[A=\begin{bmatrix} 0 & 0 & 0 & 0 […]
  • Elements of Finite Order of an Abelian Group form a SubgroupElements of Finite Order of an Abelian Group form a Subgroup Let $G$ be an abelian group and let $H$ be the subset of $G$ consisting of all elements of $G$ of finite order. That is, \[H=\{ a\in G \mid \text{the order of $a$ is finite}\}.\] Prove that $H$ is a subgroup of $G$.   Proof. Note that the identity element $e$ of […]
  • Find a Basis For the Null Space of a Given $2\times 3$ MatrixFind a Basis For the Null Space of a Given $2\times 3$ Matrix Let \[A=\begin{bmatrix} 1 & 1 & 0 \\ 1 &1 &0 \end{bmatrix}\] be a matrix. Find a basis of the null space of the matrix $A$. (Remark: a null space is also called a kernel.)   Solution. The null space $\calN(A)$ of the matrix $A$ is by […]
  • Is the Derivative Linear Transformation Diagonalizable?Is the Derivative Linear Transformation Diagonalizable? Let $\mathrm{P}_2$ denote the vector space of polynomials of degree $2$ or less, and let $T : \mathrm{P}_2 \rightarrow \mathrm{P}_2$ be the derivative linear transformation, defined by \[ T( ax^2 + bx + c ) = 2ax + b . \] Is $T$ diagonalizable? If so, find a diagonal matrix which […]

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