# triangle2018

• Hyperplane 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 $a_1x_1+a_2x_2+\cdots+a_nx_n=b,$ […]
• Differentiating Linear Transformation is Nilpotent Let $P_n$ be the vector space of all polynomials with real coefficients of degree $n$ or less. Consider the differentiation linear transformation $T: P_n\to P_n$ defined by $T\left(\, f(x) \,\right)=\frac{d}{dx}f(x).$ (a) Consider the case $n=2$. Let $B=\{1, x, x^2\}$ be a […]
• Dimension of the Sum of Two Subspaces Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$. Then prove that $\dim(U+V) \leq \dim(U)+\dim(V).$   Definition (The sum of subspaces). Recall that the sum of subspaces $U$ and $V$ is $U+V=\{\mathbf{x}+\mathbf{y} \mid […] • Subspace Spanned By Cosine and Sine Functions Let \calF[0, 2\pi] be the vector space of all real valued functions defined on the interval [0, 2\pi]. Define the map f:\R^2 \to \calF[0, 2\pi] by \[\left(\, f\left(\, \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta […] • Two Eigenvectors Corresponding to Distinct Eigenvalues are Linearly Independent Let A be an n\times n matrix. Suppose that \lambda_1, \lambda_2 are distinct eigenvalues of the matrix A and let \mathbf{v}_1, \mathbf{v}_2 be eigenvectors corresponding to \lambda_1, \lambda_2, respectively. Show that the vectors \mathbf{v}_1, \mathbf{v}_2 are […] • The Sum of Cosine Squared in an Inner Product Space Let \mathbf{v} be a vector in an inner product space V over \R. Suppose that \{\mathbf{u}_1, \dots, \mathbf{u}_n\} is an orthonormal basis of V. Let \theta_i be the angle between \mathbf{v} and \mathbf{u}_i for i=1,\dots, n. Prove that \[\cos […] • Eigenvalues and Eigenvectors of Matrix Whose Diagonal Entries are 3 and 9 Elsewhere Find all the eigenvalues and eigenvectors of the matrix \[A=\begin{bmatrix} 3 & 9 & 9 & 9 \\ 9 &3 & 9 & 9 \\ 9 & 9 & 3 & 9 \\ 9 & 9 & 9 & 3 \end{bmatrix}.$ (Harvard University, Linear Algebra Final Exam Problem)   Hint. Instead of […]
• Find the Formula for the Power of a Matrix Let $A=\begin{bmatrix} 1 & 1 & 1 \\ 0 &0 &1 \\ 0 & 0 & 1 \end{bmatrix}$ be a $3\times 3$ matrix. Then find the formula for $A^n$ for any positive integer $n$.   Proof. We first compute several powers of $A$ and guess the general formula. We […]