linear=algebra-eye-catch2

• Dual Vector Space and Dual Basis, Some Equality Let $V$ be a finite dimensional vector space over a field $k$ and let $V^*=\Hom(V, k)$ be the dual vector space of $V$. Let $\{v_i\}_{i=1}^n$ be a basis of $V$ and let $\{v^i\}_{i=1}^n$ be the dual basis of $V^*$. Then prove that $x=\sum_{i=1}^nv^i(x)v_i$ for any vector $x\in […] • Prove that any Algebraic Closed Field is Infinite Prove that any algebraic closed field is infinite. Definition. A field$F$is said to be algebraically closed if each non-constant polynomial in$F[x]$has a root in$F$. Proof. Let$F$be a finite field and consider the polynomial $f(x)=1+\prod_{a\in […] • Find Values of h so that the Given Vectors are Linearly Independent Find the value(s) of h for which the following set of vectors \[\left \{ \mathbf{v}_1=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} h \\ 1 \\ -h \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 1 \\ 2h \\ 3h+1 […] • Column Vectors of an Upper Triangular Matrix with Nonzero Diagonal Entries are Linearly Independent Suppose M is an n \times n upper-triangular matrix. If the diagonal entries of M are all non-zero, then prove that the column vectors are linearly independent. Does the conclusion hold if we do not assume that M has non-zero diagonal entries? Proof. […] • Group Homomorphism, Preimage, and Product of Groups Let G, G' be groups and let f:G \to G' be a group homomorphism. Put N=\ker(f). Then show that we have \[f^{-1}(f(H))=HN.$ Proof.$(\subset)$Take an arbitrary element$g\in f^{-1}(f(H))$. Then we have$f(g)\in f(H)$. It follows that there exists$h\in H$[…] • Find All Matrices Satisfying a Given Relation Let$a$and$b$be two distinct positive real numbers. Define matrices $A:=\begin{bmatrix} 0 & a\\ a & 0 \end{bmatrix}, \,\, B:=\begin{bmatrix} 0 & b\\ b& 0 \end{bmatrix}.$ Find all the pairs$(\lambda, X)$, where$\lambda$is a real number and$X$is a […] • Finitely Generated Torsion Module Over an Integral Domain Has a Nonzero Annihilator (a) Let$R$be an integral domain and let$M$be a finitely generated torsion$R$-module. Prove that the module$M$has a nonzero annihilator. In other words, show that there is a nonzero element$r\in R$such that$rm=0$for all$m\in M$. Here$r$does not depend on […] • Find the Inverse Linear Transformation if the Linear Transformation is an Isomorphism Let$T:\R^3 \to \R^3$be the linear transformation defined by the formula $T\left(\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \,\right)=\begin{bmatrix} x_1+3x_2-2x_3 \\ 2x_1+3x_2 \\ x_2+x_3 \end{bmatrix}.$ Determine whether$T\$ is an […]