# probability

Probability problems

• Dot 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 $A, B$ 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 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? 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 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$An$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$, denoted […] • 7 Problems on Skew-Symmetric Matrices Let$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$is […] • Any Automorphism of the Field of Real Numbers Must be the Identity Map Prove that any field automorphism of the field of real numbers$\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 […]