Euclidean Vector Spaces

Vectors are introduced—first in the plane, then in Euclidean space. Students may explore vector addition and scalar multiplication in the associated software. The dot product is introduced, along with its basic properties, including its use in computing length and angles. The Cauchy-Schwarz inequality and triangle inequality are proved, with emphasis on the relative simplicity of “coordinate-free” computation (using the properties of the dot product rather than its definition). Orthogonality of vectors and projection of one vector onto another are introduced, and the theorem that solutions to homogeneous systems are orthogonal to rows of the coefficient matrix is proved. The cross product is defined by formally using cofactor expansion. The concept of matrices representing linear transformations, and the meaning of matrix multiplication in this context, are introduced. Rotations in the plane are introduced geometrically and become a recurring example in the text, including their orthogonality shown later. The chapter concludes with the geometric interpretation of the absolute value of the determinant of a matrix as the volume of the parallelepiped spanned by the rows or columns of the matrix.