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Purdue University Linear Algebra Exam Problems and Solutions


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  • Three Equivalent Conditions for a Ring to be a FieldThree Equivalent Conditions for a Ring to be a Field Let $R$ be a ring with $1$. Prove that the following three statements are equivalent. The ring $R$ is a field. The only ideals of $R$ are $(0)$ and $R$. Let $S$ be any ring with $1$. Then any ring homomorphism $f:R \to S$ is injective.   Proof. […]
  • Find the Inverse Matrix Using the Cayley-Hamilton TheoremFind the Inverse Matrix Using the Cayley-Hamilton Theorem Find the inverse matrix of the matrix \[A=\begin{bmatrix} 1 & 1 & 2 \\ 9 &2 &0 \\ 5 & 0 & 3 \end{bmatrix}\] using the Cayley–Hamilton theorem.   Solution. To use the Cayley-Hamilton theorem, we first compute the characteristic polynomial $p(t)$ of […]
  • Are these vectors in the Nullspace of the Matrix?Are these vectors in the Nullspace of the Matrix? Let $A=\begin{bmatrix} 1 & 0 & 3 & -2 \\ 0 &3 & 1 & 1 \\ 1 & 3 & 4 & -1 \end{bmatrix}$. For each of the following vectors, determine whether the vector is in the nullspace $\calN(A)$. (a) $\begin{bmatrix} -3 \\ 0 \\ 1 \\ 0 \end{bmatrix}$ […]
  • Column Vectors of an Upper Triangular Matrix with Nonzero Diagonal Entries are Linearly IndependentColumn 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. […]
  • Is an Eigenvector of a Matrix an Eigenvector of its Inverse?Is an Eigenvector of a Matrix an Eigenvector of its Inverse? Suppose that $A$ is an $n \times n$ matrix with eigenvalue $\lambda$ and corresponding eigenvector $\mathbf{v}$. (a) If $A$ is invertible, is $\mathbf{v}$ an eigenvector of $A^{-1}$? If so, what is the corresponding eigenvalue? If not, explain why not. (b) Is $3\mathbf{v}$ an […]
  • Matrix Representations for Linear Transformations of the Vector Space of PolynomialsMatrix Representations for Linear Transformations of the Vector Space of Polynomials Let $P_2(\R)$ be the vector space over $\R$ consisting of all polynomials with real coefficients of degree $2$ or less. Let $B=\{1,x,x^2\}$ be a basis of the vector space $P_2(\R)$. For each linear transformation $T:P_2(\R) \to P_2(\R)$ defined below, find the matrix representation […]
  • $(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain.$(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain. Let $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$.   Proof. Consider the ring $R[t]$, where $t$ is a variable. Since $R$ is an integral domain, so is $R[t]$. Define the function $\Psi:R[x,y] \to R[t]$ sending […]
  • An Example of a Matrix that Cannot Be a CommutatorAn Example of a Matrix that Cannot Be a Commutator Let $I$ be the $2\times 2$ identity matrix. Then prove that $-I$ cannot be a commutator $[A, B]:=ABA^{-1}B^{-1}$ for any $2\times 2$ matrices $A$ and $B$ with determinant $1$.   Proof. Assume that $[A, B]=-I$. Then $ABA^{-1}B^{-1}=-I$ implies \[ABA^{-1}=-B. […]

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