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Probability problems

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  • Dot Product, Lengths, and Distances of Complex VectorsDot Product, Lengths, and Distances of Complex Vectors For this problem, use the complex vectors \[ \mathbf{w}_1 = \begin{bmatrix} 1 + i \\ 1 - i \\ 0 \end{bmatrix} , \, \mathbf{w}_2 = \begin{bmatrix} -i \\ 0 \\ 2 - i \end{bmatrix} , \, \mathbf{w}_3 = \begin{bmatrix} 2+i \\ 1 - 3i \\ 2i \end{bmatrix} . \] Suppose $\mathbf{w}_4$ is […]
  • If Two Matrices Have the Same Rank, Are They Row-Equivalent?If Two Matrices Have the Same Rank, Are They Row-Equivalent? If $A, B$ have the same rank, can we conclude that they are row-equivalent? If so, then prove it. If not, then provide a counterexample.   Solution. Having the same rank does not mean they are row-equivalent. For a simple counterexample, consider $A = […]
  • Inner Products, Lengths, and Distances of 3-Dimensional Real VectorsInner Products, Lengths, and Distances of 3-Dimensional Real Vectors For this problem, use the real vectors \[ \mathbf{v}_1 = \begin{bmatrix} -1 \\ 0 \\ 2 \end{bmatrix} , \mathbf{v}_2 = \begin{bmatrix} 0 \\ 2 \\ -3 \end{bmatrix} , \mathbf{v}_3 = \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} . \] Suppose that $\mathbf{v}_4$ is another vector which is […]
  • Is the Sum of a Nilpotent Matrix and an Invertible Matrix Invertible?Is the Sum of a Nilpotent Matrix and an Invertible Matrix Invertible? A square matrix $A$ is called nilpotent if some power of $A$ is the zero matrix. Namely, $A$ is nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix. Suppose that $A$ is a nilpotent matrix and let $B$ be an invertible matrix of […]
  • The Length of a Vector is Zero if and only if the Vector is the Zero VectorThe Length of a Vector is Zero if and only if the Vector is the Zero Vector Let $\mathbf{v}$ be an $n \times 1$ column vector. Prove that $\mathbf{v}^\trans \mathbf{v} = 0$ if and only if $\mathbf{v}$ is the zero vector $\mathbf{0}$.   Proof. Let $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} $. Then we […]
  • A One Side Inverse Matrix is the Inverse Matrix: If $AB=I$, then $BA=I$A One Side Inverse Matrix is the Inverse Matrix: If $AB=I$, then $BA=I$ An $n\times n$ matrix $A$ is said to be invertible if there exists an $n\times n$ matrix $B$ such that $AB=I$, and $BA=I$, where $I$ is the $n\times n$ identity matrix. If such a matrix $B$ exists, then it is known to be unique and called the inverse matrix of $A$, denoted […]
  • 7 Problems on Skew-Symmetric Matrices7 Problems on Skew-Symmetric Matrices Let $A$ and $B$ be $n\times n$ skew-symmetric matrices. Namely $A^{\trans}=-A$ and $B^{\trans}=-B$. (a) Prove that $A+B$ is skew-symmetric. (b) Prove that $cA$ is skew-symmetric for any scalar $c$. (c) Let $P$ be an $m\times n$ matrix. Prove that $P^{\trans}AP$ is […]
  • Any Automorphism of the Field of Real Numbers Must be the Identity MapAny Automorphism of the Field of Real Numbers Must be the Identity Map Prove that any field automorphism of the field of real numbers $\R$ must be the identity automorphism.   Proof. We prove the problem by proving the following sequence of claims. Let $\phi:\R \to \R$ be an automorphism of the field of real numbers […]

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