When does a vector add a new direction? Experiment with linear combinations, find a nontrivial zero relation, and follow the proof.一個向量什麼時候會帶來新方向？動手做線性組合（linear combination），找出一組不平凡的零關係（nontrivial relation），再跟著證明一步步走。
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。
All coefficients in this section are real. “Nontrivial” means at least one coefficient is nonzero; it does not mean that every coefficient must be nonzero.本節所有係數都是實數。「不平凡」（nontrivial）的意思是至少有一個係數不為 0，並不是每個係數都必須不為 0。
| Vector向量 | coordinate 1座標 1 | coordinate 2座標 2 |
|---|
Coordinates between −5 and 5; fractions such as 1/3 are accepted. Conclusions use exact arithmetic.座標範圍為 −5 到 5，可以輸入分數（例如 1/3）。結論都用精確的分數運算得出。
Try coefficients from −5 to 5, including fractions. The definition itself allows all real coefficients.係數可以在 −5 到 5 之間試，也可以用分數。定義本身允許任何實數係數。
Span means all possible linear combinations. Pale shading indicates a plane; a pale line indicates a one-dimensional span; a point indicates {0}. Overlapping arrows do not change the exact calculation.張成空間（span）是指所有可能的線性組合。淡色塗滿表示張成整個平面；淡色直線表示一維的張成空間；只有一個點表示 {0}。箭頭重疊不影響精確計算。
One nonzero vector in ℝ² is independent but spans only a line. The list (1,0),(0,1),(1,1) spans ℝ² but is dependent. A basis must be both independent and spanning.ℝ² 中一個非零向量是線性獨立的，但只張成一條直線。向量組 (1,0),(0,1),(1,1) 張成 ℝ²，卻是線性相依的。基底（basis）必須同時線性獨立，而且張成整個空間。
For u=(u₁,u₂), v=(v₁,v₂), det[u v]=u₁v₂−u₂v₁. A nonzero determinant makes the only zero relation trivial. The next section uses equations, so no determinant theory is required.對 u=(u₁,u₂)、v=(v₁,v₂)，det[u v]=u₁v₂−u₂v₁。行列式（determinant）不為 0 時，唯一的零關係就是平凡關係。下一節改用方程式，所以不需要先學行列式理論。
Add vectors one at a time and drag their tips (they snap to grid points; arrow keys work too). The pale blue region is the current span: everywhere linear combinations can reach. Watch whether each new vector makes the span bigger.一次加入一個向量，在圖上拖動箭頭尖端（會對齊格點，也可以用方向鍵）。淺藍色是目前的張成空間：所有線性組合能到達的地方。注意看每加一個向量，張成有沒有變大。
This section follows the vectors you entered in step 1. Form the matrix A with those vectors as columns. The question is whether Ac=0 has a solution other than c=0.本節沿用你在步驟 1 輸入的向量。把這些向量當成欄（column），組成矩陣 A。問題是：Ac=0 除了 c=0 之外，還有沒有其他解？
You may swap two equations, multiply an equation by a nonzero scalar, or add a multiple of another equation. The right-hand side stays zero.可以交換兩條方程式、把一條方程式乘上非零純量，或加上另一條方程式的倍數。等號右邊始終是 0。
A pivot is a leading 1 used to solve for a coefficient after elimination. A coefficient in a column without a pivot can be chosen freely; the other coefficients are then determined by the equations.主元（pivot）是消去後每一列最前面的 1，用來解出一個係數。沒有主元的欄所對應的係數可以自由選擇（自由係數，free coefficient）；其他係數再由方程式決定。
Here, the number of pivots is the rank. If every column has a pivot, all coefficients in Ac=0 are forced to zero. A free coefficient produces a nontrivial zero relation.這裡主元的個數就是秩（rank）。如果每一欄都有主元，Ac=0 的所有係數都被迫等於 0；只要有自由係數，就能得到不平凡的零關係。
A polynomial is zero only when every coefficient is zero. Enter the coefficient lists of three polynomials.多項式只有在每個係數都是 0 時才是零多項式。請輸入三個多項式的係數。
| Polynomial多項式 | constant常數項 | t | t² |
|---|
Switch the field to ℂ first: over ℝ, a = −i is not an allowed coefficient.先把體切換成 ℂ 才能使用：在 ℝ 上，a = −i 不是可以用的係數。
These are vectors in ℂ; the choice of scalar field determines which coefficient pairs you may use.這些都是 ℂ 中的向量；選擇哪個純量體，決定了你可以使用哪些係數。
For three or more vectors, “no proportional pair” does not imply independence. For example, (1,0)+(0,1)−(1,1)=0.三個以上的向量時，「兩兩不成比例」並不保證線性獨立。例如 (1,0)+(0,1)−(1,1)=0。
Allow 30–35 minutes for exploration and practice. All exercises use real coefficients unless a different field is specified. You can retry after reading the feedback.探索與練習約需 30–35 分鐘。除非題目指定其他純量體，所有練習都使用實數係數。看完回饋後可以再試。
Any valid nontrivial relation is accepted, including fractions.任何正確的不平凡關係都算對，也可以用分數。
Find a,b such that au+bv=(2,3). The feedback then shows why these coefficients are unique.求係數 a,b，使 au+bv=(2,3)。核對後會說明為什麼這組係數是唯一的。
The last implication uses independence. It proves uniqueness for every representable target, not just this one.最後一步用到線性獨立。這證明了每個可以被表示的目標向量，表示法都是唯一的，不只這一個。
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。