# Boston-college-exam-eye-catch

by Yu ·

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- Coordinate Vectors and Dimension of Subspaces (Span) Let $V$ be a vector space over $\R$ and let $B$ be a basis of $V$. Let $S=\{v_1, v_2, v_3\}$ be a set of vectors in $V$. If the coordinate vectors of these vectors with respect to the basis $B$ is given as follows, then find the dimension of $V$ and the dimension of the span of […]
- Find the Rank of a Matrix with a Parameter Find the rank of the following real matrix. \[ \begin{bmatrix} a & 1 & 2 \\ 1 &1 &1 \\ -1 & 1 & 1-a \end{bmatrix},\] where $a$ is a real number. (Kyoto University, Linear Algebra Exam) Solution. The rank is the number of nonzero rows of a […]
- The Symmetric Group is a Semi-Direct Product of the Alternating Group and a Subgroup $\langle(1,2) \rangle$ Prove that the symmetric group $S_n$, $n\geq 3$ is a semi-direct product of the alternating group $A_n$ and the subgroup $\langle(1,2) \rangle$ generated by the element $(1,2)$. Definition (Semi-Direct Product). Internal Semi-Direct-Product Recall that a group $G$ is […]
- Find the Nullspace and Range of the Linear Transformation $T(f)(x) = f(x)-f(0)$ Let $C([-1, 1])$ denote the vector space of real-valued functions on the interval $[-1, 1]$. Define the vector subspace \[W = \{ f \in C([-1, 1]) \mid f(0) = 0 \}.\] Define the map $T : C([-1, 1]) \rightarrow W$ by $T(f)(x) = f(x) - f(0)$. Determine if $T$ is a linear map. If […]
- A Linear Transformation $T: U\to V$ cannot be Injective if $\dim(U) > \dim(V)$ Let $U$ and $V$ be finite dimensional vector spaces over a scalar field $\F$. Consider a linear transformation $T:U\to V$. Prove that if $\dim(U) > \dim(V)$, then $T$ cannot be injective (one-to-one). Hints. You may use the folowing facts. A linear […]
- For Fixed Matrices $R, S$, the Matrices $RAS$ form a Subspace Let $V$ be the vector space of $k \times k$ matrices. Then for fixed matrices $R, S \in V$, define the subset $W = \{ R A S \mid A \in V \}$. Prove that $W$ is a vector subspace of $V$. Proof. We verify the subspace criteria: the zero vector of $V$ is in $W$, and […]
- Eigenvalues and Eigenvectors of The Cross Product Linear Transformation We fix a nonzero vector $\mathbf{a}$ in $\R^3$ and define a map $T:\R^3\to \R^3$ by \[T(\mathbf{v})=\mathbf{a}\times \mathbf{v}\] for all $\mathbf{v}\in \R^3$. Here the right-hand side is the cross product of $\mathbf{a}$ and $\mathbf{v}$. (a) Prove that $T:\R^3\to \R^3$ is […]
- Non-Example of a Subspace in 3-dimensional Vector Space $\R^3$ Let $S$ be the following subset of the 3-dimensional vector space $\R^3$. \[S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, x_1, x_2, x_3 \in \Z \right\}, \] where $\Z$ is the set of all integers. […]