This set of exercises is retrieved from the fifth chapter of Linear Algebra by Robert J. Valenza. Note that these solutions are not fully elaborated; You have to fill the descriptions by yourself. Problem 5.1 Solve the following matrix equation for \(x,\) \(y,\) \(z\) and \(w.\) \[ \begin{pmatrix} 1&2 \\ 0&1 \end{pmatrix} \begin{pmatrix} x&y \\ z&w \end{pmatrix} = \begin{pmatrix} 10&2 \\ 4&2 \end{pmatrix} \] Solution. Taking \(R_1 \,\leftarrow\, R_1 – 2R_2 ,\) we obtain \[\left( \begin{array}{cc|cc} 1 & 2 & …
This set of exercises is retrieved from the fourth chapter of Linear Algebra by Robert J. Valenza. Note that these solutions are not fully elaborated; You have to fill the descriptions by yourself. Problem 4.1 Let \(v_1 ,\) \(\cdots ,\) \(v_n \) be linearly independent family in a vector space \(V.\) Show that if \(i\ne j,\) then \(v_i \ne v_j .\) In other words, a linearly independent family can never contain a repeated vector. Solution. Suppose not, that is, suppose …
This set of exercises is retrieved from the third chapter of Linear Algebra by Robert J. Valenza. Note that these solutions are not fully elaborated; You have to fill the descriptions by yourself. Problem 3.1 Show that the solution set \(W\) of vectors \((x_1 ,\,x_2 )\) in \(\mathbb{R}^2\) satisfying the equation \[x_1 + 8x_2 = 0\] is a subspace of \(\mathbb{R}^2 .\) Solution. The solution set is \[S = \left\{ (-8s,\, s) \,\vert\, s\in\mathbb{R} \right\}.\] This set is closed under …
This set of exercises is retrieved from the second chapter of Linear Algebra by Robert J. Valenza. Note that these solutions are not fully elaborated; You have to fill the descriptions by yourself. Problem 2.1 Give an example of a noncommutative group of \(24\) elements. Solution. \(S_4 .\) Problem 2.2 Give an example of a group \(G\) and a nonempty subset \(H\) of \(G\) which is closed under the operation defined on \(G,\) but is not a subgroup of \(G.\) …
This set of exercises is retrieved from the second chapter of Linear Algebra by Robert J. Valenza. Note that these solutions are not fully elaborated; You have to fill the descriptions by yourself. Problem 1.1 Find the sets \(S,\) \(T\) and \(U\) and functions \(f: S \rightarrow T\) and \(g: T \rightarrow U\) such that \(g \circ f\) is injective, but \(g\) is not injective. Solution. Take \(S = U = \left\{ 1 \right\} ,\,\, T = \left\{ 0,\,1 \right\},\) …
이 포스트에서는 직사각형 영역에서 정의된 함수의 이중적분을 정의하고, 연속함수의 적분 가능성을 증명합니다. 리만 적분의 엄밀한 정의가 기억나지 않는다면 일변수 함수의 리만 적분을 소개하는 이전 글(바로가기)을 먼저 읽어 보기 바랍니다. 리만 적분의 정의 먼저 이중적분을 정의하자. \(I = [a,\,b]\)와 \(J = [c,\,d]\)가 길이가 양수인 구간이고 \(R = I \times J\)라고 하자. 그리고 \[\begin{gather} P_I = \left\{ x_0 ,\, x_1 ,\, x_2 ,\, \cdots ,\, x_m \right\} , \tag{1}\\[7pt] P_J = \left\{ y_0 ,\, y_1 ,\, y_2 ,\, \cdots ,\, y_n \right\} …