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  • Determine Whether Given Subsets in $\R^4$ are Subspaces or NotDetermine Whether Given Subsets in $\R^4$ are Subspaces or Not (a) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying \[2x+4y+3z+7w+1=0.\] Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a […]
  • Find a Condition that a Vector be a Linear CombinationFind a Condition that a Vector be a Linear Combination Let \[\mathbf{v}=\begin{bmatrix} a \\ b \\ c \end{bmatrix}, \qquad \mathbf{v}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix} 2 \\ -1 \\ 2 \end{bmatrix}.\] Find the necessary and […]
  • Find All Symmetric Matrices satisfying the EquationFind All Symmetric Matrices satisfying the Equation Find all $2\times 2$ symmetric matrices $A$ satisfying $A\begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$? Express your solution using free variable(s).   Solution. Let $A=\begin{bmatrix} a & b\\ c& d \end{bmatrix}$ […]
  • Automorphism Group of $\Q(\sqrt[3]{2})$ Over $\Q$.Automorphism Group of $\Q(\sqrt[3]{2})$ Over $\Q$. Determine the automorphism group of $\Q(\sqrt[3]{2})$ over $\Q$. Proof. Let $\sigma \in \Aut(\Q(\sqrt[3]{2}/\Q)$ be an automorphism of $\Q(\sqrt[3]{2})$ over $\Q$. Then $\sigma$ is determined by the value $\sigma(\sqrt[3]{2})$ since any element $\alpha$ of $\Q(\sqrt[3]{2})$ […]
  • If a Matrix $A$ is Full Rank, then $\rref(A)$ is the Identity MatrixIf a Matrix $A$ is Full Rank, then $\rref(A)$ is the Identity Matrix Prove that if $A$ is an $n \times n$ matrix with rank $n$, then $\rref(A)$ is the identity matrix. Here $\rref(A)$ is the matrix in reduced row echelon form that is row equivalent to the matrix $A$.   Proof. Because $A$ has rank $n$, we know that the $n \times n$ […]
  • Solve the System of Linear Equations Using the Inverse Matrix of the Coefficient MatrixSolve the System of Linear Equations Using the Inverse Matrix of the Coefficient Matrix Consider the following system of linear equations \begin{align*} 2x+3y+z&=-1\\ 3x+3y+z&=1\\ 2x+4y+z&=-2. \end{align*} (a) Find the coefficient matrix $A$ for this system. (b) Find the inverse matrix of the coefficient matrix found in (a) (c) Solve the system using […]
  • A Singular Matrix and Matrix Equations $A\mathbf{x}=\mathbf{e}_i$ With Unit VectorsA Singular Matrix and Matrix Equations $A\mathbf{x}=\mathbf{e}_i$ With Unit Vectors Let $A$ be a singular $n\times n$ matrix. Let \[\mathbf{e}_1=\begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix}, \dots, […]
  • The Inverse Matrix is UniqueThe Inverse Matrix is Unique Let $A$ be an $n\times n$ invertible matrix. Prove that the inverse matrix of $A$ is uniques.   Hint. That the inverse matrix of $A$ is unique means that there is only one inverse matrix of $A$. (That's why we say "the" inverse matrix of $A$ and denote it by […]

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