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Purdue University Linear Algebra Exam Problems and Solutions


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  • Idempotent Elements and Zero Divisors in a Ring and in an Integral DomainIdempotent Elements and Zero Divisors in a Ring and in an Integral Domain Prove the following statements. (a) If $a\neq 1$ is an idempotent element of $R$, then $a$ is a zero divisor. (b) Suppose that $R$ is an integral domain. Determine all the idempotent elements of $R$.   Definitions (Idempotent, Zero Divisor, Integral […]
  • Fundamental Theorem of Finitely Generated Abelian Groups and its applicationFundamental Theorem of Finitely Generated Abelian Groups and its application In this post, we study the Fundamental Theorem of Finitely Generated Abelian Groups, and as an application we solve the following problem. Problem. Let $G$ be a finite abelian group of order $n$. If $n$ is the product of distinct prime numbers, then prove that $G$ is isomorphic […]
  • Calculate Determinants of MatricesCalculate Determinants of Matrices Calculate the determinants of the following $n\times n$ matrices. \[A=\begin{bmatrix} 1 & 0 & 0 & \dots & 0 & 0 &1 \\ 1 & 1 & 0 & \dots & 0 & 0 & 0 \\ 0 & 1 & 1 & \dots & 0 & 0 & 0 \\ \vdots & \vdots […]
  • Any Vector is a Linear Combination of Basis Vectors UniquelyAny Vector is a Linear Combination of Basis Vectors Uniquely Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a basis for a vector space $V$ over a scalar field $K$. Then show that any vector $\mathbf{v}\in V$ can be written uniquely as \[\mathbf{v}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3,\] where $c_1, c_2, c_3$ are […]
  • How to Find a Formula of the Power of a MatrixHow to Find a Formula of the Power of a Matrix Let $A= \begin{bmatrix} 1 & 2\\ 2& 1 \end{bmatrix}$. Compute $A^n$ for any $n \in \N$. Plan. We diagonalize the matrix $A$ and use this Problem. Steps. Find eigenvalues and eigenvectors of the matrix $A$. Diagonalize the matrix $A$. Use […]
  • Find an Orthonormal Basis of the Given Two Dimensional Vector SpaceFind an Orthonormal Basis of the Given Two Dimensional Vector Space Let $W$ be a subspace of $\R^4$ with a basis \[\left\{\, \begin{bmatrix} 1 \\ 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \\ 1 \end{bmatrix} \,\right\}.\] Find an orthonormal basis of $W$. (The Ohio State […]
  • Linear Transformation $T(X)=AX-XA$ and Determinant of Matrix RepresentationLinear Transformation $T(X)=AX-XA$ and Determinant of Matrix Representation Let $V$ be the vector space of all $n\times n$ real matrices. Let us fix a matrix $A\in V$. Define a map $T: V\to V$ by \[ T(X)=AX-XA\] for each $X\in V$. (a) Prove that $T:V\to V$ is a linear transformation. (b) Let $B$ be a basis of $V$. Let $P$ be the matrix […]
  • Given the Characteristic Polynomial, Find the Rank of the MatrixGiven the Characteristic Polynomial, Find the Rank of the Matrix Let $A$ be a square matrix and its characteristic polynomial is give by \[p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).\] Find the rank of $A$. (The Ohio State University, Linear Algebra Final Exam Problem)   Solution. Note that the degree of the characteristic polynomial […]

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