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4: Rⁿ

Last updated

Apr 3, 2025
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- 3.4E: Exercises for Section 3.4
- 4.1: Review of Vectors

Fatemeh Yarahmadi, De Anza College
Brigham Young University via Lyryx

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4.1: Review of Vectors
This page covers the foundational concepts of vectors in $\mathbb{R}^n$ , including position vectors, vector operations (addition, subtraction, scalar multiplication), and geometric interpretations. It explains distance between points using the Pythagorean theorem, introduces unit vectors and their calculation, and discusses linear combinations of vectors.
- 4.1E: Exercises for Section 4.1
4.2: Dot and Cross Product
There are two ways of multiplying vectors which are of great importance in applications. The first of these is called the dot product. When we take the dot product of vectors, the result is a scalar. For this reason, the dot product is also called the scalar product and sometimes the inner product.
- 4.2E: Exercises for Section 4.2
4.3: Lines and Planes
We can use the concept of vectors and points to find equations for arbitrary lines in $\mathbb{R}^n$ , although in this section the focus will be on lines in $\mathbb{R}^3$ .
- 4.3E: Exercises for Section 4.3
4.4: Spanning Sets in Rⁿ
By generating all linear combinations of a set of vectors one can obtain various subsets of $\mathbb{R}^{n}$ which we call subspaces. For example what set of vectors in $\mathbb{R}^{3}$ generate the $XY$ -plane? What is the smallest such set of vectors can you find? The tools of spanning, linear independence and basis are exactly what is needed to answer these and similar questions and are the focus of this section.
- 4.4.E: Exercise for Section 4.4
4.5: Linear Independence
This section discusses the linear dependence and independence between vectors.
- 4.5.E: Exercise for Section 4.5
4.6: Subspaces and Bases
The goal of this section is to develop an understanding of a subspace of $\mathbb{R}^n$ .
- 4.6.E: Exercise for Section 4.6
4.7: Row, Column and Null Spaces
This section discusses the Row, Column, and Null Spaces of a matrix, focusing on their definitions, properties, and computational methods.
- 4.7.E: Exercise for Section 4.7
4.8: Orthogonal Vectors and Matrices
In this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions for the span of a set of vectors and a linear independent set of vectors.
- 4.8.E: Exercise for Section 4.8
4.9: Gram-Schmidt Process
The Gram-Schmidt process is an algorithm to transform a set of vectors into an orthonormal set spanning the same subspace, that is generating the same collection of linear combinations.
- 4.9.E: Exercises for Section 4.9
4.10: Orthogonal Projections
An important use of the Gram-Schmidt Process is in orthogonal projections, the focus of this section.
- 4.10.E: Exercise for Section 4.10
4.11: Least Squares Approximation
In this section, we discuss a very important technique derived from orthogonal projections: the least squares approximation.
- 4.11.E: Exercises for Section 4.11

Thumbnail: Animation showing how the vector cross product (green) varies when the angles between the blue and red vectors is changed. (Public Domain; Nicostella via Wikipedia)

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