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Johns Hopkins Linear Algebra Exam Problems and Solutions


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  • Find a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or LessFind a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or Less Let $\calP_3$ be the vector space of all polynomials of degree $3$ or less. Let \[S=\{p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} p_1(x)&=1+3x+2x^2-x^3 & p_2(x)&=x+x^3\\ p_3(x)&=x+x^2-x^3 & p_4(x)&=3+8x+8x^3. \end{align*} (a) […]
  • Matrices Satisfying the Relation $HE-EH=2E$Matrices Satisfying the Relation $HE-EH=2E$ Let $H$ and $E$ be $n \times n$ matrices satisfying the relation \[HE-EH=2E.\] Let $\lambda$ be an eigenvalue of the matrix $H$ such that the real part of $\lambda$ is the largest among the eigenvalues of $H$. Let $\mathbf{x}$ be an eigenvector corresponding to $\lambda$. Then […]
  • Properties of Nonsingular and Singular MatricesProperties of Nonsingular and Singular Matrices An $n \times n$ matrix $A$ is called nonsingular if the only solution of the equation $A \mathbf{x}=\mathbf{0}$ is the zero vector $\mathbf{x}=\mathbf{0}$. Otherwise $A$ is called singular. (a) Show that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is […]
  • Every Diagonalizable Nilpotent Matrix is the Zero MatrixEvery Diagonalizable Nilpotent Matrix is the Zero Matrix Prove that if $A$ is a diagonalizable nilpotent matrix, then $A$ is the zero matrix $O$.   Definition (Nilpotent Matrix) A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$. Proof. Main Part Since $A$ is […]
  • Give a Formula For a Linear Transformation From $\R^2$ to $\R^3$Give a Formula For a Linear Transformation From $\R^2$ to $\R^3$ Let $\{\mathbf{v}_1, \mathbf{v}_2\}$ be a basis of the vector space $\R^2$, where \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 1 \end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix} 1 \\ -1 \end{bmatrix}.\] The action of a linear transformation $T:\R^2\to \R^3$ on the […]
  • Solving a System of Linear Equations By Using an Inverse MatrixSolving a System of Linear Equations By Using an Inverse Matrix Consider the system of linear equations \begin{align*} x_1&= 2, \\ -2x_1 + x_2 &= 3, \\ 5x_1-4x_2 +x_3 &= 2 \end{align*} (a) Find the coefficient matrix and its inverse matrix. (b) Using the inverse matrix, solve the system of linear equations. (The Ohio […]
  • Find a Nonsingular Matrix $A$ satisfying $3A=A^2+AB$Find a Nonsingular Matrix $A$ satisfying $3A=A^2+AB$ (a) Find a $3\times 3$ nonsingular matrix $A$ satisfying $3A=A^2+AB$, where \[B=\begin{bmatrix} 2 & 0 & -1 \\ 0 &2 &-1 \\ -1 & 0 & 1 \end{bmatrix}.\] (b) Find the inverse matrix of $A$.   Solution (a) Find a $3\times 3$ nonsingular matrix $A$. Assume […]
  • All the Eigenvectors of a Matrix Are Eigenvectors of Another MatrixAll the Eigenvectors of a Matrix Are Eigenvectors of Another Matrix Let $A$ and $B$ be an $n \times n$ matrices. Suppose that all the eigenvalues of $A$ are distinct and the matrices $A$ and $B$ commute, that is $AB=BA$. Then prove that each eigenvector of $A$ is an eigenvector of $B$. (It could be that each eigenvector is an eigenvector for […]

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