## 如何學好線性代數？

Q1. 二階行列式定義為 $\begin{vmatrix} a&b\\ c&d \end{vmatrix}=ad-bc$，為甚麼不定義為 $\begin{vmatrix} a&b\\ c&d \end{vmatrix}=bc-ad$

Q2. 一個 $2\times 2$ 階矩陣 $\begin{bmatrix} a&b\\ c&d \end{bmatrix}$ 的行列式是平面上兩個向量 $\begin{bmatrix} a\\ c \end{bmatrix}$$\begin{bmatrix} b\\ d \end{bmatrix}$$(a,b)$$(c,d)$ 所張平行四邊形的 (有號) 面積。三維空間的兩個向量 $\begin{bmatrix} a\\ c\\ e \end{bmatrix}$$\begin{bmatrix} b\\ d\\ f \end{bmatrix}$ 也張開一平行四邊形，我們何不定義 $3\times 2$ 階矩陣 $\begin{bmatrix} a&b\\ c&d\\ e&f \end{bmatrix}$ 的「行列式」為該平行四邊形的面積？

Q3. 怎麼解釋 $\begin{vmatrix} a&b\\ c&d \end{vmatrix}=\begin{vmatrix} a&b\\ 2a+c&2b+d \end{vmatrix}$，但 $\begin{vmatrix} a&b\\ c&d \end{vmatrix}\neq\begin{vmatrix} a&b\\ a+2c&b+2d \end{vmatrix}$

Q4. 為甚麼兩個向量 $\mathbf{x}=(x_1,x_2,x_3)$$\mathbf{y}=(y_1,y_2,y_3)$ 沒有乘法運算卻有外積 (cross product)？譬如，為甚麼不定義向量乘法 $\mathbf{x}\times\mathbf{y}=(x_1y_1,x_2y_2,x_3y_3)$

Q5. 如何理解一個矩陣的最大線性獨立的行向量數 (行秩，column rank) 等於最大線性獨立的列向量數 (列秩，row rank)？

Q6. 為甚麼 $2\times 2$ 階矩陣形成的集合可稱為向量空間？既然平面上向量是一個具有方向與長度的數學物件，如何理解矩陣 $\begin{bmatrix} a&b\\ c&d \end{bmatrix}$ 的方向與長度？我們需要引入甚麼必要的運算？

Q7. 行列式可乘公式 $\det(AB)=(\det A)(\det B)$，即兩個同階方陣乘積的行列式為等於這兩個方陣的行列式的乘積，這個事實的幾何意義是甚麼？

Q8. 矩陣乘法不具有交換律，為甚麼不定義一種矩陣乘法使得同階方陣的乘積具有交換律？

Q9. 「線性」是甚麼意思？為甚麼向量空間也稱為線性空間？對於向量 $\mathbf{x},\mathbf{y}$ 與純量 $\alpha$，線性變換 $T$ 滿足 $T(\mathbf{x}+\mathbf{y})=T(\mathbf{x})+T(\mathbf{y})$$T(\alpha\mathbf{x})=\alpha T(\mathbf{x})$，何以具備這兩個性質就稱為線性變換？

Q10. 為甚麼線性變換的定義域與到達域都限定為向量空間 (或子空間) 而非任意的向量集合？

Q11. 向量空間的一個子空間為甚麼一定要包含零向量？為甚麼 $ax+by=0$ 的解集合稱為子空間，但 $ax+by+c=0$$c\neq 0$，的解集合卻不稱為子空間？

Q12. 一個線性變換可以用不同的矩陣來表示，那麼不同的線性變換可以用相同的矩陣來表示嗎？

Q13. 為甚麼線性代數課本都沒有討論如何解矩陣方程，譬如，滿足 $X^2=I$ 以及 $Y^2=Y$$2\times 2$ 階矩陣 $X$$Y$ 要怎麼解？

[1] Paul R. Halmos, I Want to Be a Mathematician, 1985, pp 40-41. 原文：“The algebra course was hard and I worked at it furiously;…When I say furiously, I mean furiously. Brahana didn’t know how to be clear, the text was Bôcher’s book (which I thought was mess), and my dominant emotion during much of the time that I spent on the subject was exasperation reaching to anger….somehow I survive my introduction to linear algebra. I didn’t really begin to understand what the subject was about till four or five years later, after I got my Ph.D. and heard von Neumann talk about operator theory.”
[2] 原文：“Any fool can make a rule, and any fool will mind it.”
[3] 英譯文：“It is only possible to understand the commutativity of multiplication by counting and re-counting soldiers by ranks and files or by calculating the area of a rectangle in the two ways. Any attempt to do without this interference by physics and reality into mathematics is sectarianism and isolationism which destroy the image of mathematics as a useful human activity in the eyes of all sensible people.”
[4] 原文：“Brick walls are there for a reason: they let us prove how badly we want things.”
[5] 原文：“The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.”
[6] 引用自維基百科：無伴奏大提琴組曲
[7] 原文：“Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?”