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所在平台: Udemy |
课程主页: https://www.udemy.com/course/inner-product-spaces/
课程评论：没有评论
**课程概述：内积空间** 本课程将深入探讨“内积空间”，主要围绕实数向量空间及其内积运算展开。我们将从二维和三维空间中的向量（点积和叉积）入手，逐步推广到更高维度的实数空间 $R^n$ 。 **核心内容包括：** * **点积与向量的几何性质：** 学习 $R^n$ 中点积的定义及其在衡量向量长度、角度和正交性方面的应用。 * **正交补与正交基：** 掌握子空间的正交补概念，并学习如何构建子空间的正交基，这是理解更复杂结构的关键。 * **正交投影：** 探索向量向二维子空间进行正交投影的原理和计算方法。 * **Gram-Schmidt 正交化过程：** 学习如何通过给定的基来生成子空间的正交基，并应用这一过程进行矩阵的 QR 分解。 * **最小二乘法：** 利用正交投影的思想，解决实际中的最小二乘问题，寻找最优拟合解。 * **傅里叶级数：** 运用内积空间的理论，初步了解傅里叶级数的构建和应用。 **重要提示：** 本课程内容与本平台上的“线性代数”课程高度重叠。如果您已修读或计划修读“线性代数”，则无需再学习本课程。
An inner product space (over the real numbers) is a vector space with an operation called an inner product. A typical example of an inner product space of dimension n is the Euclidean space R^n with the usual dot product as the inner product (for a = (a_1, a_2,..., a_n) and b = (b_1, b_2,..., b_n) in R^n, the inner product of a and b is defined to be = a_1b_1 + a_2b_2 +.+ a_nb_n). We will concentrate on R^n and generalize the results to a general inner product space. We will proceed as follows. First, we give a discussion on vectors in R^2 and R^3 and define the dot product and the cross product of two vectors, then we discuss orthogonal complements and orthogonal basis for a subspace of R^n. Next, we consider the orthogonal projection of a vector onto a dimension two subspace. We also introduce the Gram-Schmidt process to produce an orthogonal basis for a subspace using a given basis of that subspace. Using this process we can give a QR factorization of an invertible matrix. Using the idea of the orthogonal projection of a vector onto a subspace, we form the least-squares problem and provide solutions for such problems. We also use the same idea to study the Fourier series.The topics presented in this course are included in my course "Linear Algebra" posted on this site. So if you have taken, or are going to take, "Linear Algebra", you don't need to take this course.
