This set of exercises is retrieved from the eighth 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 8.1 Using the recursive definition given in the proof of the existence of determinant, systematically evaluate the determinant of the following matrix: \[A=\begin{pmatrix}1&2&1\\0&1&1\\1&0&2\end{pmatrix}.\] Solution. \[\begin{aligned} \det (A) &= 1 \cdot …
This set of exercises is retrieved from the seventh 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 7.1 In \(\mathbb{R}^3,\) compute the inner product of \((1,\,2,\,-1)\) and \((2,\,1,\,4).\) What is the length of each vector? What is the angle between these vectors? Solution. The lengths of given …
This set of exercises is retrieved from the sixth 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 6.1 Let \(T:\mathbb{R}^2 \rightarrow \mathbb{R}\) be a linear transformation and suppose that \(T(1,\,1)=5\) and \(T(0,\,1)=2.\) Find \(T(x_1,\,x_2)\) for all \(x_1,\) \(x_2 \in \mathbb{R}.\) Solution. Suppose \((x_1 ,\,x_2 )\) be given. …
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 \(w.\) \[ \begin{pmatrix} 1&2 \\ 0&1 \end{pmatrix} \begin{pmatrix} x&y \\ z&w \end{pmatrix} = \begin{pmatrix} 10&2 \\ 4&2 \end{pmatrix} \] …
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, …
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 …
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 \(G\) …
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 …
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 …
‘자기주도적 학습 과제’는 스스로 공부하는 학생들에게 학습의 방향을 안내해주기 위한 문제입니다. 매주 5문제가 제공됩니다. Thomas Calculus 관련 단원을 공부한 후 충분히 생각하면서 문제를 풀어보세요. 여러분의 실력 향상에 도움이 될 것입니다. **** **** **** 9주차 9주차 문제의 관련 단원은 2.5, 10.2, 14.1절입니다. 다음 문제에서 \(D\)는 \(\mathbb{R}^2\)의 부분집합을 나타냅니다. \(D\)가 닫힌집합이라고 합시다. 또한 수열 \(\left\{ \textbf{x}_n \right\}\)의 모든 점이 \(D\)에 속한다고 합시다. 만약 \(\left\{ \textbf{x}_n \right\}\)이 …