UC-Berkeley-eye-catch

LoadingAdd to solve later

Linear Algebra exam problems and solutions at University of California, Berkeley


LoadingAdd to solve later

More from my site

  • Beautiful Formulas for pi=3.14…Beautiful Formulas for pi=3.14… The number $\pi$ is defined a s the ratio of a circle's circumference $C$ to its diameter $d$: \[\pi=\frac{C}{d}.\] $\pi$ in decimal starts with 3.14... and never end. I will show you several beautiful formulas for $\pi$.   Art Museum of formulas for $\pi$ […]
  • Determine the Quotient Ring $\Z[\sqrt{10}]/(2, \sqrt{10})$Determine the Quotient Ring $\Z[\sqrt{10}]/(2, \sqrt{10})$ Let \[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}\] be an ideal of the ring \[\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}.\] Then determine the quotient ring $\Z[\sqrt{10}]/P$. Is $P$ a prime ideal? Is $P$ a maximal ideal?   Solution. We […]
  • Row Equivalence of Matrices is TransitiveRow Equivalence of Matrices is Transitive If $A, B, C$ are three $m \times n$ matrices such that $A$ is row-equivalent to $B$ and $B$ is row-equivalent to $C$, then can we conclude that $A$ is row-equivalent to $C$? If so, then prove it. If not, then provide a counterexample.   Definition (Row […]
  • Prove that the Length $\|A^n\mathbf{v}\|$ is As Small As We Like.Prove that the Length $\|A^n\mathbf{v}\|$ is As Small As We Like. Consider the matrix \[A=\begin{bmatrix} 3/2 & 2\\ -1& -3/2 \end{bmatrix} \in M_{2\times 2}(\R).\] (a) Find the eigenvalues and corresponding eigenvectors of $A$. (b) Show that for $\mathbf{v}=\begin{bmatrix} 1 \\ 0 \end{bmatrix}\in \R^2$, we can choose […]
  • Prove the Ring Isomorphism $R[x,y]/(x) \cong R[y]$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 […]
  • Orthogonality of Eigenvectors of a Symmetric Matrix Corresponding to Distinct EigenvaluesOrthogonality of Eigenvectors of a Symmetric Matrix Corresponding to Distinct Eigenvalues Suppose that a real symmetric matrix $A$ has two distinct eigenvalues $\alpha$ and $\beta$. Show that any eigenvector corresponding to $\alpha$ is orthogonal to any eigenvector corresponding to $\beta$. (Nagoya University, Linear Algebra Final Exam Problem)   Hint. Two […]
  • Find a Polynomial Satisfying the Given Conditions on DerivativesFind a Polynomial Satisfying the Given Conditions on Derivatives Find a cubic polynomial \[p(x)=a+bx+cx^2+dx^3\] such that $p(1)=1, p'(1)=5, p(-1)=3$, and $ p'(-1)=1$.   Solution. By differentiating $p(x)$, we obtain \[p'(x)=b+2cx+3dx^2.\] Thus the given conditions are […]
  • Common Eigenvector of Two Matrices $A, B$ is Eigenvector of $A+B$ and $AB$.Common Eigenvector of Two Matrices $A, B$ is Eigenvector of $A+B$ and $AB$. Let $\lambda$ be an eigenvalue of $n\times n$ matrices $A$ and $B$ corresponding to the same eigenvector $\mathbf{x}$. (a) Show that $2\lambda$ is an eigenvalue of $A+B$ corresponding to $\mathbf{x}$. (b) Show that $\lambda^2$ is an eigenvalue of $AB$ corresponding to […]

Leave a Reply

Your email address will not be published. Required fields are marked *