\(n\)이 \(2\) 이상인 자연수이고 \(a_1 ,\) \(\cdots ,\) \(a_n ,\) \(b_1 ,\) \(\cdots,\) \(b_n\)이 모두 실수일 때 다음이 성립한다. \[\left( \sum_{i=1}^n a_i b_i \right)^2 \le \left( \sum_{i=1}^n a_i ^2\right) \left(\sum_{i=1}^n b_i ^2 \right).\tag{1}\] 이 부등식을 코시-슈바르츠 부등식(Cauchy-Schwarz inequality)이라고 부른다. 라그랑주 승수법(method of Lagrange’s multiplier)을 이용하여 이 부등식을 증명해 보자. 증명을 마칠 때까지 첨수 \(i\)와 \(j\)는 \(n\) 이하의 자연수를 나타내는 것으로 약속한다. 증명 과정은 두 단계로 진행된다. 첫째 단계에서는 길이가 \(1\)인 두 벡터 \(\mathbb{x}=(x_i)\)와 \(\mathbb{y}=(y_i)\)에 대하여 \[-1 \le \mathbb{x}\cdot\mathbb{y} \le 1 …
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 \det \left[\begin{array}{cc} 1 & 1 \\ 0 & 2 \end{array}\right] – 2 \cdot \det \left[\begin{array}{cc} 0 …
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 vectors are \[\begin{aligned} \lVert (1,\,2,\,-1)\rVert &= \sqrt{1^2 + 2^2 + (-1)^2} = \sqrt{6} ,\\[4pt] \lVert (2,\,1,\,4)\rVert …
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. Take \(\lambda_1 = x_1 ,\) \(\lambda_2 = x_2 – x_1 ,\) then \[(x_1 ,\,x_2 ) = …
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 …