Vectors

This chapter systematically studies the basic concepts of vectors, linear combinations, linear dependence, vector spaces, bases and dimension, inner product, and orthogonalization.

Basic Concepts of Vectors

• n-dimensional vector: $\vec{a} = (a_1, a_2, \dots, a_n)$
• Vector addition, scalar multiplication

Linear Combination and Linear Dependence

• Linear combination: $k_1\vec{a}_1 + k_2\vec{a}_2 + \dots + k_m\vec{a}_m$
• Linear dependence: there exist coefficients, not all zero, such that the combination is zero
• Linear independence: only when all coefficients are zero does the combination equal zero

Maximal Linearly Independent Set and Rank

• Maximal linearly independent set: the largest linearly independent subset of a set of vectors
• Rank of a set of vectors: the number of vectors in a maximal linearly independent set

Vector Spaces, Bases, and Dimension

• Vector space: a set closed under addition and scalar multiplication, satisfying certain rules
• Basis: a linearly independent set that can uniquely represent any vector in the space
• Dimension: the number of vectors in a basis

Coordinate Transformation and Transition Matrix

• Definition and computation of coordinate transformation and transition matrix

Inner Product and Orthogonalization

• Inner product: $\vec{a} \cdot \vec{b} = \sum a_i b_i$
• Schmidt orthogonalization method
• Orthogonal basis, orthogonal matrix

Exercises

1. Determine the linear dependence of the set $\{(1,2,3), (2,4,6), (1,0,1)\}$.
2. Find the standard basis of $\mathbb{R}^3$.
3. Let $\vec{a} = (1,2,3), \vec{b} = (2,1,0)$, compute $\vec{a} \cdot \vec{b}$.
4. Use the Schmidt orthogonalization method to orthogonalize $\vec{a}_1 = (1,1,0), \vec{a}_2 = (1,0,1)$.
5. Find the rank of the set $\{(1,2,3), (2,4,6), (3,6,9)\}$.