# Stanford-university-exam-eye-catch

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- Hyperplane Through Origin is Subspace of 4-Dimensional Vector Space Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying \[2x+3y+5z+7w=0.\] Then prove that the set $S$ is a subspace of $\R^4$. (Linear Algebra Exam Problem, The Ohio State […]
- Finite Group and Subgroup Criteria Let $G$ be a finite group and let $H$ be a subset of $G$ such that for any $a,b \in H$, $ab\in H$. Then show that $H$ is a subgroup of $G$. Proof. Let $a \in H$. To show that $H$ is a subgroup of $G$, it suffices to show that the inverse $a^{-1}$ is in $H$. If […]
- Quiz 13 (Part 1) Diagonalize a Matrix Let \[A=\begin{bmatrix} 2 & -1 & -1 \\ -1 &2 &-1 \\ -1 & -1 & 2 \end{bmatrix}.\] Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$. That is, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that […]
- Three Linearly Independent Vectors in $\R^3$ Form a Basis. Three Vectors Spanning $\R^3$ Form a Basis. Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of three-dimensional vectors in $\R^3$. (a) Prove that if the set $B$ is linearly independent, then $B$ is a basis of the vector space $\R^3$. (b) Prove that if the set $B$ spans $\R^3$, then $B$ is a basis of […]
- Condition that Two Matrices are Row Equivalent We say that two $m\times n$ matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations. Let $A$ and $I$ be $2\times 2$ matrices defined as follows. \[A=\begin{bmatrix} 1 & b\\ c& d \end{bmatrix}, \qquad […]
- Orthogonal Nonzero Vectors Are Linearly Independent Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ be a set of nonzero vectors in $\R^n$. Suppose that $S$ is an orthogonal set. (a) Show that $S$ is linearly independent. (b) If $k=n$, then prove that $S$ is a basis for $\R^n$. Proof. (a) […]
- Find all Column Vector $\mathbf{w}$ such that $\mathbf{v}\mathbf{w}=0$ for a Fixed Vector $\mathbf{v}$ Let $\mathbf{v} = \begin{bmatrix} 2 & -5 & -1 \end{bmatrix}$. Find all $3 \times 1$ column vectors $\mathbf{w}$ such that $\mathbf{v} \mathbf{w} = 0$. Solution. Let $\mathbf{w} = \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix}$. Then we want \[\mathbf{v} […]
- Algebraic Number is an Eigenvalue of Matrix with Rational Entries A complex number $z$ is called algebraic number (respectively, algebraic integer) if $z$ is a root of a monic polynomial with rational (respectively, integer) coefficients. Prove that $z \in \C$ is an algebraic number (resp. algebraic integer) if and only if $z$ is an eigenvalue of […]