# 1. 线性代数与矩阵

n维线性空间中n个线性无关的向量可以构成该线性空间的一组基，选定了一组基，集合中的每个元素(即向量)就可以被唯一表示了。想像下，欧几里德的三维空间，选定了坐标系(如[1,0,0]、[0,1,0]、[0,0,1])，空间内的任一个点都可以被唯一表示成(a,b,c)。

When you talk about matrices, you’re allowed to talk about things like the entry in the 3rd row and 4th column, and so forth. In this setting, matrices are useful for representing things like transition probabilities in a Markov chain, where each entry indicates the probability of transitioning from one state to another.

In linear algebra, however, you instead talk about linear transformations, which are not a list of numbers, although sometimes it is convenient to use a particular matrix to write down a linear transformation. However, when you’re given a linear transformation, you’re not allowed to ask for things like the entry in its 3rd row and 4th column because questions like these depend on a choice of basis. Instead, you’re only allowed to ask for things that don’t depend on the basis, such as the rank, the trace, the determinant, or the set of eigenvalues. This point of view may seem unnecessarily restrictive, but it is fundamental to a deeper understanding of pure mathematics.

Additivity:              f(x + y) = f(x) + f(y).
Homogeneity of degree 1: fx) = αf(x) for all α.

# 2. 从线性方程到矩阵

3. 线性变换

3.1 标量乘法

Homogeneity of degree 1: fx) = αf(x) for all α.

3.2 加法

Additivity:              f(x + y) = f(x) + f(y).

• 加法结合律：u + (v + w) = (u + v) + w
• 加法的单位元：V存在零向量的0，∀ v ∈ V , v + 0 = v
• 加法的逆元素：∀v∈V, ∃w∈V，使得 v + w = 0
• 加法交换律：v + w = w + v

3.3 矩阵乘法

http://blog.stata.com/2011/03/03/understanding-matrices-intuitively-part-1/

translation, but no rotation, scaling, or skew.

http://math.stackexchange.com/questions/31725/intuition-behind-matrix-multiplication

BetterExplained: An Intuitive Guide to Linear Algebra

1. Matrix multiputation

BetterExplained: An Intuitive Guide to Linear Algebra

This article offers a geometric explanation of singular value decompositions and look at some of the applications of them.

[1]WolframMathworld: Linear Transformation