# graphs-of-characteristic-polynomials

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• Linear Transformation and a Basis of the Vector Space $\R^3$ Let $T$ be a linear transformation from the vector space $\R^3$ to $\R^3$. Suppose that $k=3$ is the smallest positive integer such that $T^k=\mathbf{0}$ (the zero linear transformation) and suppose that we have $\mathbf{x}\in \R^3$ such that $T^2\mathbf{x}\neq \mathbf{0}$. Show […]
• Every Basis of a Subspace Has the Same Number of Vectors Let $V$ be a subspace of $\R^n$. Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ is a basis of the subspace $V$. Prove that every basis of $V$ consists of $k$ vectors in $V$.   Hint. You may use the following fact: Fact. If […]
• A Matrix Representation of a Linear Transformation and Related Subspaces Let $T:\R^4 \to \R^3$ be a linear transformation defined by $T\left (\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \,\right) = \begin{bmatrix} x_1+2x_2+3x_3-x_4 \\ 3x_1+5x_2+8x_3-2x_4 \\ x_1+x_2+2x_3 \end{bmatrix}.$ (a) Find a matrix $A$ such that […]
• Restriction of a Linear Transformation on the x-z Plane is a Linear Transformation Let $T:\R^3 \to \R^3$ be a linear transformation and suppose that its matrix representation with respect to the standard basis is given by the matrix $A=\begin{bmatrix} 1 & 0 & 2 \\ 0 &3 &0 \\ 4 & 0 & 5 \end{bmatrix}.$ (a) Prove that the linear transformation […]
• Find a Linear Transformation Whose Image (Range) is a Given Subspace Let $V$ be the subspace of $\R^4$ defined by the equation $x_1-x_2+2x_3+6x_4=0.$ Find a linear transformation $T$ from $\R^3$ to $\R^4$ such that the null space $\calN(T)=\{\mathbf{0}\}$ and the range $\calR(T)=V$. Describe $T$ by its matrix […]
• A Line is a Subspace if and only if its $y$-Intercept is Zero Let $\R^2$ be the $x$-$y$-plane. Then $\R^2$ is a vector space. A line $\ell \subset \mathbb{R}^2$ with slope $m$ and $y$-intercept $b$ is defined by $\ell = \{ (x, y) \in \mathbb{R}^2 \mid y = mx + b \} .$ Prove that $\ell$ is a subspace of $\mathbb{R}^2$ if and only if $b = […] • Inner Product, Norm, and Orthogonal Vectors Let$\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3$are vectors in$\R^n$. Suppose that vectors$\mathbf{u}_1$,$\mathbf{u}_2$are orthogonal and the norm of$\mathbf{u}_2$is$4$and$\mathbf{u}_2^{\trans}\mathbf{u}_3=7$. Find the value of the real number$a$in […] • Injective Group Homomorphism that does not have Inverse Homomorphism Let$A=B=\Z$be the additive group of integers. Define a map$\phi: A\to B$by sending$n$to$2n$for any integer$n\in A$. (a) Prove that$\phi$is a group homomorphism. (b) Prove that$\phi$is injective. (c) Prove that there does not exist a group homomorphism$\psi:B […]