linear=algebra-eye-catch2

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  • Find the Inverse Linear Transformation if the Linear Transformation is an IsomorphismFind the Inverse Linear Transformation if the Linear Transformation is an Isomorphism Let $T:\R^3 \to \R^3$ be the linear transformation defined by the formula \[T\left(\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \,\right)=\begin{bmatrix} x_1+3x_2-2x_3 \\ 2x_1+3x_2 \\ x_2-x_3 \end{bmatrix}.\] Determine whether $T$ is an […]
  • Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a RelationQuiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation (a) Find the inverse matrix of \[A=\begin{bmatrix} 1 & 0 & 1 \\ 1 &0 &0 \\ 2 & 1 & 1 \end{bmatrix}\] if it exists. If you think there is no inverse matrix of $A$, then give a reason. (b) Find a nonsingular $2\times 2$ matrix $A$ such that \[A^3=A^2B-3A^2,\] where […]
  • Quiz 7. Find a Basis of the Range, Rank, and Nullity of a MatrixQuiz 7. Find a Basis of the Range, Rank, and Nullity of a Matrix (a) Let $A=\begin{bmatrix} 1 & 3 & 0 & 0 \\ 1 &3 & 1 & 2 \\ 1 & 3 & 1 & 2 \end{bmatrix}$. Find a basis for the range $\calR(A)$ of $A$ that consists of columns of $A$. (b) Find the rank and nullity of the matrix $A$ in part (a).   Solution. (a) […]
  • Example of an Infinite Group Whose Elements Have Finite OrdersExample of an Infinite Group Whose Elements Have Finite Orders Is it possible that each element of an infinite group has a finite order? If so, give an example. Otherwise, prove the non-existence of such a group.   Solution. We give an example of a group of infinite order each of whose elements has a finite order. Consider […]
  • Probability that Alice Wins n Games Before Bob Wins m GamesProbability that Alice Wins n Games Before Bob Wins m Games Alice and Bob play some game against each other. The probability that Alice wins one game is $p$. Assume that each game is independent. If Alice wins $n$ games before Bob wins $m$ games, then Alice becomes the champion of the game. What is the probability that Alice becomes the […]
  • Differentiating Linear Transformation is NilpotentDifferentiating Linear Transformation is Nilpotent Let $P_n$ be the vector space of all polynomials with real coefficients of degree $n$ or less. Consider the differentiation linear transformation $T: P_n\to P_n$ defined by \[T\left(\, f(x) \,\right)=\frac{d}{dx}f(x).\] (a) Consider the case $n=2$. Let $B=\{1, x, x^2\}$ be a […]
  • If the Order is an Even Perfect Number, then a Group is not SimpleIf the Order is an Even Perfect Number, then a Group is not Simple (a) Show that if a group $G$ has the following order, then it is not simple. $28$ $496$ $8128$ (b) Show that if the order of a group $G$ is equal to an even perfect number then the group is not simple. Hint. Use Sylow's theorem. (See the post Sylow’s Theorem […]
  • Find the Inverse Matrices if Matrices are Invertible by Elementary Row OperationsFind the Inverse Matrices if Matrices are Invertible by Elementary Row Operations For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix. (a) $A=\begin{bmatrix} 1 & 3 & -2 \\ 2 &3 &0 \\ […]

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