Chapter 3 · Foundation第 3 章 · 基礎

Matrix Transformation Lab矩陣即線性變換

A matrix is a function: it sends every vector to a new vector. Watch the whole grid move, read where e₁ and e₂ land, and let the determinant tell you about area and flips.矩陣是一個函數：把每個向量送到新的向量。看整張方格網怎麼變形、讀出 e₁ 和 e₂ 被送到哪裡，再讓行列式（determinant）告訴你面積和翻面的事。

First experiment先做這三步
  1. Press Reset and check that A=I=[1 0; 0 1]: the grid stays put, blue e₁ points to (1,0) and green e₂ to (0,1).按「重設」，確認 A=I=[1 0; 0 1]：方格不動，藍色 e₁ 指向 (1,0)，綠色 e₂ 指向 (0,1)。
  2. Predict first: after changing the top-right entry b from 0 to 1, where do e₁ and e₂ go? Record your prediction, then change b.先預測：把右上角 b 從 0 改成 1 之後，e₁ 和 e₂ 各會被送到哪裡？記錄預測後再改 b。
  3. Check the calculation: the image of v=(2,1) is Av=2·(1,0)+1·(1,1)=(3,1). Find the tip of the red arrow, then see why the grid leans: e₂ now goes to (1,1).核對算式：v=(2,1) 的像 Av=2·(1,0)+1·(1,1)=(3,1)。在圖上找到紅色箭頭的終點，再看看方格為什麼變斜：e₂ 被送到 (1,1)。

Predict → change one value → inspect the calculation → read the key idea.先預測 → 改一個值 → 檢查算式 → 看懂重點概念。

A matrix is a function: vector in, vector out.矩陣是一個函數：輸入向量，輸出向量。

The 2×2 matrix A sends each vector v=(x,y) to Av. Instead of moving one arrow, look at the whole grid at once: every grid line moves together.2×2 矩陣 A 把每個向量 v=(x,y) 送到 Av。不要只看一支箭頭，一次看整張方格網：所有格線會一起移動。

Predict, then set the matrix先預測，再設定矩陣
A =
[a b; c d]

Blue frame: column 1 (a,c). Green frame: column 2 (b,d). Entries from −5 to 5; fractions such as 1/2 are accepted.藍框是第 1 欄 (a,c)，綠框是第 2 欄 (b,d)。元素範圍 −5 到 5，可輸入分數（例如 1/2）。

The grid before and after變形前後的方格網

Pale grey: the original grid. Pale blue: the image of every grid line. The shaded parallelogram is the image of the unit square; the small flag shows whether the plane has been flipped.淡灰色是原本的方格；淺藍色是每條格線的像。上色的平行四邊形是單位正方形的像；小旗子用來看平面有沒有翻面。

Why does the origin never move?為什麼原點永遠不動？
A(0,0) = (a·0+b·0, c·0+d·0) = (0,0).

This holds for every matrix A, not only the ones you try. Grid lines also stay straight, parallel and evenly spaced, because A(u+v)=Au+Av and A(cv)=c·Av.這對每一個矩陣 A 都成立，不只是你試過的例子。格線也一直保持直線、平行、等距，因為 A(u+v)=Au+Av、A(cv)=c·Av。

Two ways to read AvAv 的兩種讀法
Av = (a·x + b·y, c·x + d·y) = x·(a, c) + y·(b, d).
Row view: each output coordinate uses one row. Column view: Av is a linear combination of the two columns, exactly like au+bv in Chapter 1.列的讀法：每個輸出座標用一列（row）。欄的讀法：Av 是兩欄（column）的線性組合，和第 1 章的 au+bv 一模一樣。
Explore first; the next section explains why the columns decide everything.先動手探索；下一節說明為什麼兩欄就決定了一切。
Drag a tip, change a column拖箭頭就是改一欄Drag the blue tip first and watch a and c change together; then drag the green tip and watch b and d. In the challenge you may drag the tips or type the matrix. Remember: a row runs across, a column runs down.先拖藍色箭頭，看 a、c 兩格一起變；再拖綠色箭頭，看 b、d 一起變。挑戰中可以拖箭頭，也可以直接輸入矩陣。記得：列（row）是橫的，欄（column）是直的。

Ae₁ is column 1; Ae₂ is column 2.Ae₁ 是第 1 欄，Ae₂ 是第 2 欄。

The standard basis vectors are e₁=(1,0) and e₂=(0,1). Every vector is a combination of them, so once you know where they land, you know where everything lands.標準基底（standard basis）是 e₁=(1,0) 和 e₂=(0,1)。每個向量都是它們的組合，所以只要知道它們被送到哪裡，就知道所有向量被送到哪裡。

From the definition從定義出發
A e₁ = A(1,0) = (a·1+b·0, c·1+d·0) = (a, c)
A e₂ = A(0,1) = (a·0+b·1, c·0+d·1) = (b, d)
A v = A(x e₁ + y e₂) = x·Ae₁ + y·Ae₂

So the matrix is simply a list of two images written as columns. The last line is Chapter 1’s linear combination au+bv, with u=Ae₁, v=Ae₂, a=x and b=y.所以矩陣就是把兩個像寫成兩欄。最後一行就是第 1 章的線性組合 au+bv，只是 u=Ae₁、v=Ae₂、a=x、b=y。

Common slip常見錯誤Ae₁ is the first column (a,c), not the first row (a,b). A row runs across; a column runs down.Ae₁ 是第 1 欄 (a,c)，不是第 1 列 (a,b)。列（row）是橫的，欄（column）是直的。
Looking ahead to Chapter 4預告第 4 章

The Transformer’s W is also a list of columns.Transformer 的 W 也是一排欄。

In Chapter 4, W has 3 rows and 2 columns: it sends a vector in ℝ² to three scores in ℝ³. The same rule holds:第 4 章的 W 有 3 列 2 欄：把 ℝ² 的向量送到 ℝ³ 的三個分數。規則完全一樣：

Wx = x₁·(column 1 of W) + x₂·(column 2 of W).

So if x₂ increases by 1, every score increases by the matching number in column 2. Keep this in mind when you reach the Transformer lab.所以 x₂ 增加 1 時，每個分數都會加上第 2 欄中對應的數。到 Transformer 實驗室時可以用上。

Challenge挑戰

讓方格對上影子Match the shadow

每一關有一個淡紫色的「影子」：某個矩陣把單位正方形和小旗子送到的位置。拖動藍色 Ae₁ 和綠色 Ae₂ 的箭頭尖端，或直接輸入矩陣，讓你的像和影子完全重合。拖箭頭時注意：藍色箭頭只會改變 a、c，綠色只會改變 b、d。Each level shows a pale violet “shadow”: where some matrix sends the unit square and the small flag. Drag the tips of blue Ae₁ and green Ae₂, or type the matrix, until your image matches the shadow exactly. Notice that dragging blue changes only a and c, and green only b and d.

A =
Guess the area first先猜面積再看行列式Estimate how large the image of the unit square is before reading the determinant. A negative det still means a positive area; the sign only says the plane flipped. When det=0, recall Chapter 2: the columns are dependent.先估計單位正方形的像有多大，再看行列式（determinant）。det 是負數時面積仍然是正的，負號只代表翻面。det=0 時，回想第 2 章：兩欄相依。

det A tells you how much area grows, and whether the plane flips.det A 告訴你面積放大幾倍，以及平面有沒有翻面。

The unit square has area 1. Its image is the parallelogram spanned by the two columns. The determinant det A = ad − bc measures that parallelogram with a sign.單位正方形的面積是 1，它的像是兩欄張成的平行四邊形。行列式 det A = ad − bc 用「帶正負號」的方式量這個平行四邊形。

Try a matrix試一個矩陣
A =
Reading the determinant怎麼讀行列式
area of image = |det A| × original area
det > 0: same orientation. det < 0: flipped, like a mirror image. det = 0: the plane is squashed onto a line or a point.det > 0：方向不變。det < 0：翻面，像照鏡子。det = 0：平面被壓成一條線或一個點。
Unit square, flag and turning direction單位正方形、小旗子與旋轉方向

The curved arrow turns from Ae₁ to Ae₂. Counterclockwise means no flip; clockwise means the plane has flipped. The flag turns into its mirror image when det < 0.弧形箭頭從 Ae₁ 轉向 Ae₂：逆時針表示沒有翻面，順時針表示翻面了。det < 0 時，小旗子會變成鏡像。

Back to Chapter 2回到第 2 章

Dependent columns squash the plane.相依的兩欄會把平面壓扁。

In Chapter 2, (1,2) and (2,4) were dependent: 2·(1,2) − 1·(2,4) = (0,0). Use them as the columns of A:第 2 章中，(1,2) 和 (2,4) 相依：2·(1,2) − 1·(2,4) = (0,0)。把它們當成 A 的兩欄：

A = [1 2; 2 4],   det A = 1·4 − 2·2 = 0.
Av = x·(1,2) + y·(2,4) = (x+2y)·(1,2).
A(2,−1) = 2·(1,2) − 1·(2,4) = (0,0).

Every output lies on the line y=2x, the span from Chapter 2. The nonzero vector (2,−1) is sent to 0, just like (0,0). Two different inputs give the same output, so the output cannot tell you which input you started from: A cannot be undone.每個輸出都落在直線 y=2x 上，也就是第 2 章的張成空間。非零向量 (2,−1) 被送到 0，和 (0,0) 一樣。兩個不同輸入得到同一個輸出，所以只看輸出無法知道原本的輸入：A 無法還原。

Surprises worth a pause值得停下來想的例子

Shear [1 1; 0 1]: the square leans over, but det = 1·1 − 1·0 = 1, so the area is unchanged.剪切（shear）[1 1; 0 1]：正方形歪了，但 det = 1·1 − 1·0 = 1，面積沒有變。

Reflection [0 1; 1 0]: e₁ and e₂ swap places. Area is still 1, but det = −1: the flag is mirrored.鏡射（reflection）[0 1; 1 0]：e₁ 和 e₂ 互換位置。面積仍是 1，但 det = −1：小旗子變成鏡像。

Rotation by 90° [0 −1; 1 0]: det = 0·0 − (−1)·1 = 1. Turning is not flipping.旋轉 90° [0 −1; 1 0]：det = 0·0 − (−1)·1 = 1。旋轉不是翻面。

Optional: why is |ad − bc| an area?選讀：為什麼 |ad − bc| 是面積？

Take the case a, b, c, d > 0 with column 1 below column 2. The parallelogram sits in a rectangle of size (a+b)×(c+d). Remove two triangles of area ac/2, two triangles of area bd/2 and two rectangles of area bc:考慮 a、b、c、d > 0 且第 1 欄在第 2 欄下方的情形。平行四邊形放在 (a+b)×(c+d) 的長方形裡，扣掉兩個面積 ac/2 的三角形、兩個面積 bd/2 的三角形和兩個面積 bc 的長方形：

(a+b)(c+d) − ac − bd − 2bc = ad − bc.

This is one case, not a general proof; the general statement needs properties of determinants.這只是一種情形的推導，不是一般證明；一般情形需要行列式的性質。

Predict before checking先自己想，再檢查Do one question at a time. Choose an answer or enter numbers, then press Check. If you are wrong, read the explanation, edit your answer and try again. Nothing is marked against you.一次做一題。選答案或填入數值，再按檢查。答錯時看中文說明，修改後可以再試；沒有扣分。

Predict it. Check it. Learn the key idea.先預測、再核對、看懂重點。

Allow 25–30 minutes for exploration and practice. Every computed answer is checked exactly; feedback names the row, column or determinant step to revisit. Retrying is encouraged.預留 25–30 分鐘探索與練習。所有計算題都用精確分數核對；回饋會指出該回頭看的列、欄或行列式步驟。鼓勵你多試幾次。

07 · Compute Av07 · 矩陣乘向量

Let A = [2 −1; 1 3] and v = (3,2). Find Av.設 A = [2 −1; 1 3]、v = (3,2)，求 Av 的兩個座標。

08 · Build a collapsing matrix08 · 造一個壓扁矩陣

Column 1 of A is fixed at (2,−1). Choose column 2 so that A squashes the whole plane onto a line, then give a nonzero v with Av = 0.A 的第 1 欄固定為 (2,−1)。選第 2 欄，讓 A 把整個平面壓成一條線；再給一個非零向量 v，使 Av = 0。

A = [2 b; −1 d]

Many answers are possible.答案不只一種。

Keep evidence: one Av computed both by rows and by columns, one determinant with its area and orientation, and a sentence linking det = 0 to dependent columns from Chapter 2.留下學習紀錄：一個分別用「列」和「欄」算出的 Av、一個行列式與它代表的面積和方向，以及一句話說明 det = 0 和第 2 章「兩欄相依」的關係。
Continue to the Transformer lab →繼續學習 Transformer →
Glossary: each term in one sentence術語表：用一句話理解
Vector向量（Vector）
可做加法與純量乘法的物件。例：(2,1) 有兩個座標；多項式也可以是向量。
An object you can add and multiply by a scalar. (2,1) has two coordinates; a polynomial can also be a vector.
Scalar純量（Scalar）
拿來乘向量的數。例如 3(1,2)=(3,6)，其中 3 是純量。
A number that multiplies a vector. In 3(1,2)=(3,6), the scalar is 3.
Coordinate座標（Coordinate）
描述向量的一個數值。(2,1) 的第一座標是 2，第二座標是 1。
One of the numbers describing a vector. For (2,1), the first coordinate is 2 and the second is 1.
Coefficient係數（Coefficient）
線性組合中乘在向量前面的純量。2u−v 的係數是 2 和 −1。
The scalar in front of each vector in a linear combination. In 2u−v the coefficients are 2 and −1.
Linear combination線性組合（Linear combination）
先把向量各自乘上純量，再相加：au+bv。
Multiply each vector by a scalar, then add the results: au+bv.
Symbols符號怎麼讀（Symbols）
∈：屬於；∉：不屬於；∀：每一個；∃：至少存在一個；⇒：如果前者成立，則後者成立。ℝ 是實數集合，ℂ 是複數集合。
∈: is in; ∉: is not in; ∀: for every; ∃: there is at least one; ⇒: if the first part holds, so does the second. ℝ is the set of real numbers and ℂ the set of complex numbers.
Linear transformation線性變換（Linear transformation）
把向量送到向量的函數，而且保留加法與純量乘法：A(u+v)=Au+Av、A(cv)=cAv。2×2 矩陣就是一個例子。
A function from vectors to vectors that preserves addition and scaling: A(u+v)=Au+Av and A(cv)=cAv. Every 2×2 matrix gives one.
Matrix / row / column矩陣／列／欄（Matrix / row / column）
數字排成的長方形表格。列（row）是橫的一排，欄（column）是直的一排。[a b; c d] 的第 1 欄是 (a,c)。
A rectangular table of numbers. A row runs across; a column runs down. The first column of [a b; c d] is (a,c).
Standard basis標準基底（Standard basis）
e₁=(1,0) 與 e₂=(0,1)。每個向量都能寫成 (x,y)=x e₁+y e₂。
e₁=(1,0) and e₂=(0,1). Every vector can be written as (x,y)=x e₁+y e₂.
Image像（Image）
輸入經過函數後的輸出。Ae₁ 是 e₁ 的像，也就是 A 的第 1 欄。
The output a function gives for an input. Ae₁ is the image of e₁, which is the first column of A.
Determinant行列式（Determinant）
det [a b; c d]=ad−bc。絕對值是面積放大的倍率，正負號表示方向有沒有翻轉。
det [a b; c d]=ad−bc. Its absolute value is the area scale factor; its sign tells whether orientation flips.
Orientation方向（定向）（Orientation）
從 e₁ 轉到 e₂ 是逆時針。若像 Ae₁ 轉到 Ae₂ 變成順時針，就是翻面（det<0），像照鏡子一樣。
Turning from e₁ to e₂ is counterclockwise. If turning from Ae₁ to Ae₂ is clockwise, the plane has flipped (det<0), like a mirror image.
Shear剪切（Shear）
把方格往一側推斜，例如 [1 1; 0 1]。形狀改變，但面積不變（det=1）。
Pushes the grid sideways, as in [1 1; 0 1]. The shape changes but the area does not (det=1).
Reflection / rotation鏡射／旋轉（Reflection / rotation）
鏡射像照鏡子，det=−1，例如 [0 1; 1 0]。旋轉 90° 是 [0 −1; 1 0]，det=1。
A reflection mirrors the plane (det=−1), e.g. [0 1; 1 0]. A 90° rotation is [0 −1; 1 0], with det=1.
Collapse壓扁（Collapse）
兩欄相依時 det=0，整個平面被壓到一條直線（或一點）上，有非零向量被送到 0。
When the columns are dependent, det=0: the whole plane is squashed onto a line (or a point), and some nonzero vector is sent to 0.
Invertible可逆（Invertible）
能從輸出唯一找回輸入。2×2 矩陣可逆的條件是 det≠0。
The input can be recovered uniquely from the output. A 2×2 matrix is invertible exactly when det≠0.

Some terms are advanced; you do not need to memorise them all at once. Experiment first, then use examples to understand the definitions.有些名詞屬於進階內容，不必一次全部背起來。先操作，再用例子理解定義。