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- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/04%3A_Vector_Geometry
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/12%3A_Appendices/12.02%3A_ProofsWe must show that \(p \Rightarrow q\) where \(p\) is the statement “\(2^n - 1\) is a prime”, and \(q\) is the statement “\(n\) is a prime.” Suppose that \(p\) is true but \(q\) is false so that \(n\) ...We must show that \(p \Rightarrow q\) where \(p\) is the statement “\(2^n - 1\) is a prime”, and \(q\) is the statement “\(n\) is a prime.” Suppose that \(p\) is true but \(q\) is false so that \(n\) is not a prime, say \(n = ab\) where \(a \geq 2\) and \(b \geq 2\) are integers.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/00%3A_Front_Matter
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/02%3A_Matrix_AlgebraIn the study of systems of linear equations in Chapter [chap:1], we found it convenient to manipulate the augmented matrix of the system. In addition to originating matrix theory and the theory of det...In the study of systems of linear equations in Chapter [chap:1], we found it convenient to manipulate the augmented matrix of the system. In addition to originating matrix theory and the theory of determinants, he did fundamental work in group theory, in higher-dimensional geometry, and in the theory of invariants.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/10%3A_Inner_Product_Spaces
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/06%3A_Vector_Spaces/6.05%3A_An_Application_to_PolynomialsHere the numerator is the product of all the terms \((x - a_{0}), (x - a_{1}), \dots, (x - a_{n})\) with \((x - a_{k})\) omitted, and a similar remark applies to the denominator. \[\begin{aligned} \de...Here the numerator is the product of all the terms \((x - a_{0}), (x - a_{1}), \dots, (x - a_{n})\) with \((x - a_{k})\) omitted, and a similar remark applies to the denominator. \[\begin{aligned} \delta_0 & = \frac{(x - 0)(x - 1)}{(-1 - 0)(-1 - 1)} = \frac{1}{2}(x^2 - x) \\ \delta_1 & = \frac{(x + 1)(x - 1)}{( 0 + 1)( 0 - 1)} = -(x^2 - 1) \\ \delta_2 & = \frac{(x + 1)(x - 0)}{( 1 + 1)( 1 - 0)} = \frac{1}{2}(x^2 + x)\end{aligned} \nonumber \]
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/05%3A_Vector_Space_R/5.05%3A_Similarity_and_DiagonalizationHence the eigenvalues are \(\lambda_{1} = i\) and \(\lambda_{2} = -i\), with corresponding eigenvectors \(\mathbf{x}_1 = \left[ \begin{array}{r} 1 \\ -i \end{array} \right]\) and \(\mathbf{x}_2 = \lef...Hence the eigenvalues are \(\lambda_{1} = i\) and \(\lambda_{2} = -i\), with corresponding eigenvectors \(\mathbf{x}_1 = \left[ \begin{array}{r} 1 \\ -i \end{array} \right]\) and \(\mathbf{x}_2 = \left[ \begin{array}{r} 1 \\ i \end{array} \right].\) Hence \(A\) is diagonalizable by the complex version of Theorem [thm:016145], and the complex version of Theorem [thm:016068] shows that \(P = \left[ \begin{array}{cc} \mathbf{x}_1\ & \mathbf{x}_2 \end{array} \right]= \left[ \begin{array}{rr} 1 & 1 …
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/05%3A_Vector_Space_R/5.02%3A_Independence_and_DimensionSome spanning sets are better than others. Our interest here is in spanning sets where each vector in U has exactly one representation as a linear combination of these vectors.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/04%3A_Vector_Geometry/4.05%3A_An_Application_to_Computer_GraphicsOn the other hand, we can rotate the letter about the origin through \(\frac{\pi}{6}\) (or \(30^\circ\)) by multiplying by the matrix \(R_{\frac{\pi}{2}} = \left[ \def\arraystretch{1.5}\begin{array}{r...On the other hand, we can rotate the letter about the origin through \(\frac{\pi}{6}\) (or \(30^\circ\)) by multiplying by the matrix \(R_{\frac{\pi}{2}} = \left[ \def\arraystretch{1.5}\begin{array}{rr} \cos(\frac{\pi}{6}) & -\sin(\frac{\pi}{6})\\ \sin(\frac{\pi}{6}) & \cos(\frac{\pi}{6}) \end{array} \right] = \left[ \begin{array}{ll} 0.866 & -0.5\\ 0.5 & 0.866 \end{array} \right]\).
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/04%3A_Vector_Geometry/4.02%3A_Projections_and_PlanesAny student of geometry soon realizes that the notion of perpendicular lines is fundamental.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/02%3A_Matrix_Algebra/2.08%3A_An_Application_to_Input-Output_Economic_ModelsThus, the pricing must be such that the total output of the farming industry has the same value as the total output of the garment industry, whereas the total value of the housing industry must be \(\...Thus, the pricing must be such that the total output of the farming industry has the same value as the total output of the garment industry, whereas the total value of the housing industry must be \(\frac{3}{2}\) as much. If \(p_{i}\) and \(e_{ij}\) are as before, the value of the annual demand for product \(i\) by the producing industries themselves is \(e_{i1}p_{1} + e_{i2}p_{2} + \cdots + e_{in}p_{n}\), so the total annual revenue \(p_{i}\) of industry \(i\) breaks down as follows:
