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

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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 $\{(1,2,3), (2,4,6), (3,6,9)\}$.

Reference Answers

1. Determine linear dependence

$(2,4,6) = 2\times(1,2,3)$, so they are linearly dependent.

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2. Standard basis of $\mathbb{R}^3$

$(1,0,0), (0,1,0), (0,0,1)$

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3. $\vec{a} \cdot \vec{b}$

$1\times2 + 2\times1 + 3\times0 = 2 + 2 + 0 = 4$

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4. Schmidt orthogonalization

$\vec{b}_1 = (1,1,0)$

$\vec{b}_2 = (1,0,1) - \frac{(1,0,1)\cdot(1,1,0)}{(1,1,0)\cdot(1,1,0)}(1,1,0) = (1,0,1) - \frac{1}{2}(1,1,0) = (0.5,-0.5,1)$

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5. Rank

The three vectors are linearly dependent, so the rank is 1.

上一章节 Systems of Linear Equations Understand the structure of solutions for homogeneous and nonhomogeneous systems, master Cramer's rule, fundamental solution sets, general solutions, and elementary row transformations. 下一章节 Eigenvalues and Eigenvectors Understand the concepts of eigenvalues, eigenvectors, similarity, diagonalization, and real symmetric matrices.