# finalOSUfindeigenvalues

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• Give a Formula for a Linear Transformation if the Values on Basis Vectors are Known Let $T: \R^2 \to \R^2$ be a linear transformation. Let $\mathbf{u}=\begin{bmatrix} 1 \\ 2 \end{bmatrix}, \mathbf{v}=\begin{bmatrix} 3 \\ 5 \end{bmatrix}$ be 2-dimensional vectors. Suppose that \begin{align*} T(\mathbf{u})&=T\left( \begin{bmatrix} 1 \\ […]
• Rank and Nullity of a Matrix, Nullity of Transpose Let $A$ be an $m\times n$ matrix. The nullspace of $A$ is denoted by $\calN(A)$. The dimension of the nullspace of $A$ is called the nullity of $A$. Prove the followings. (a) $\calN(A)=\calN(A^{\trans}A)$. (b) $\rk(A)=\rk(A^{\trans}A)$.   Hint. For part (b), […]
• Prove the Ring Isomorphism $R[x,y]/(x) \cong R[y]$ Let $R$ be a commutative ring. Consider the polynomial ring $R[x,y]$ in two variables $x, y$. Let $(x)$ be the principal ideal of $R[x,y]$ generated by $x$. Prove that $R[x, y]/(x)$ is isomorphic to $R[y]$ as a ring.   Proof. Define the map $\psi: R[x,y] \to […] • True or False: If$A, B$are 2 by 2 Matrices such that$(AB)^2=O$, then$(BA)^2=O$Let$A$and$B$be$2\times 2$matrices such that$(AB)^2=O$, where$O$is the$2\times 2$zero matrix. Determine whether$(BA)^2$must be$O$as well. If so, prove it. If not, give a counter example. Proof. It is true that the matrix$(BA)^2$must be the zero […] • Find the Vector Form Solution to the Matrix Equation$A\mathbf{x}=\mathbf{0}$Find the vector form solution$\mathbf{x}$of the equation$A\mathbf{x}=\mathbf{0}$, where$A=\begin{bmatrix} 1 & 1 & 1 & 1 &2 \\ 1 & 2 & 4 & 0 & 5 \\ 3 & 2 & 0 & 5 & 2 \\ \end{bmatrix}$. Also, find two linearly independent vectors$\mathbf{x}$satisfying […] • Ring Homomorphisms and Radical Ideals Let$R$and$R'$be commutative rings and let$f:R\to R'$be a ring homomorphism. Let$I$and$I'$be ideals of$R$and$R'$, respectively. (a) Prove that$f(\sqrt{I}\,) \subset \sqrt{f(I)}$. (b) Prove that$\sqrt{f^{-1}(I')}=f^{-1}(\sqrt{I'})$(c) Suppose that$f$is […] • 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$h\in H$[…] • For What Values of$a$, Is the Matrix Nonsingular? Determine the values of a real number$a\$ such that the matrix $A=\begin{bmatrix} 3 & 0 & a \\ 2 &3 &0 \\ 0 & 18a & a+1 \end{bmatrix}$ is nonsingular.   Solution. We apply elementary row operations and obtain: \begin{align*} A=\begin{bmatrix} 3 & 0 & a […]