線性代數 Quiz3
3-5~4.2
1
1. Ture or False
(a). A zero vector space is an empty set.
F
由單一物件 0 組成
𝑉= 𝟎
(See 4.1 定義1 & 例題1)
2
1. Ture or False
(b). If W is a subspace of 𝑅3 containing a plane and a
point not in the plane, then 𝑊 = 𝑅3 .
T
𝑅3 子空間
 𝟎
 通過原點的直線
 通過原點的平面
 𝑅3
3
1. Ture or False
(c). The set 𝑅2 is a subspace of 𝑅3 .
T
𝑅3 子空間
 𝟎
 通過原點的直線
 通過原點的平面
 𝑅3
4
1. Ture or False
(d). If u, v and w are vectors in 𝑅3 , 𝐮 ∙ (𝐯 × 𝐰) = 0 iff the 3
vectors lie in the same plane.
T
定理3.5.5：
若 u、v、w 為 𝑅3 上之向量，且有共同起始點，則三向量共
平面，若
𝑢1 𝑢2 𝑢3
𝐮 ∙ 𝐯 × 𝐰 = 𝑣1 𝑣2 𝑣3 = 0
𝑤1 𝑤2 𝑤3
平行六面體無體積
or 共平面上三向量中的任兩向量外積與剩餘向量正交
5
1. Ture or False
(e). If u, v and w are vectors in 𝑅3 , where u is a
nonzero vector and 𝐮 × 𝐯 = 𝐮 × 𝐰, then 𝐯 = 𝐰.
F
讓 𝐮 = 0, 0, 1 , 𝐯 = 0, 0, 0 , 𝐰 = 0, 0, 1
𝐮 × 𝐯 = 0, 0, 0
𝐮 × 𝐰 = 0, 0, 0
但是 𝐯 ≠ 𝐰
6
1. Ture or False
(f). The set of all 3 × 3 triangular matrices equipped
with the standard operations is a vector space.
T
3 × 3 矩陣所成的集合 𝑀33 ，在平常的矩陣加法、
純量乘法運算下是一個向量空間，三角矩陣所成
集合為 𝑀33 的子空間 ，亦為一個向量空間
參考 4.1 例題4 & 4.2 例題5
7
2.
If 𝐮 ∙ 𝐯 × 𝐰 = 6, find
(a). 𝐮 𝐰 ∙ 𝐯
𝑢1 𝑢2
𝐮 ∙ 𝐰 × 𝐯 = 𝑤1 𝑤2
𝑣1 𝑣2
𝑢1 𝑢2
𝐮 ∙ 𝐯 × 𝐰 = 𝑣1 𝑣2
𝑤1 𝑤2
⇒ 𝐮 𝐰 ∙ 𝐯 = −6
𝑢3
𝑤3
𝑣3
𝑢3
𝑣3
𝑤3
change(2)(3)row
8
2.
If 𝐮 ∙ 𝐯 × 𝐰 = 6, find
(b). 𝐰 𝐯 ∙ 𝐮
𝑤1 𝑤2
𝐰 ∙ 𝐯 × 𝐮 = 𝑣1 𝑣2
𝑢1 𝑢2
𝑢1 𝑢2
𝐮 ∙ 𝐯 × 𝐰 = 𝑣1 𝑣2
𝑤1 𝑤2
⇒ 𝐰 𝐯 ∙ 𝐮 = −6
𝑤3
𝑣3
𝑢3
𝑢3
𝑣3
𝑤3
change(1)(3)row
9
2.
If 𝐮 ∙ 𝐯 × 𝐰 = 6, find
(c). 𝐯 𝐰 ∙ 𝐮
𝑣1 𝑣2
𝐯 ∙ 𝐰 × 𝐮 = 𝑤1 𝑤2
𝑢1 𝑢2
𝑤1 𝑤2
𝐰 ∙ 𝐯 × 𝐮 = 𝑣1 𝑣2
𝑢1 𝑢2
⇒𝐰 𝐯∙𝐮 =6
𝑣3
𝑤3
𝑢3
𝑤3
𝑣3
𝑢3
change(1)(2)row
10
3.
Let V be the set of all ordered pairs of real numbers, and
consider the following addition and scalar multiplication
operations on 𝐮 = 𝑢1 , 𝑢2 and 𝐯 = 𝑣1 , 𝑣2 :
𝐮 + 𝐯 = 𝑢1 + 𝑣1 − 1, 𝑢2 + 𝑣2 − 1 , 𝑘𝐮 = 𝑘𝑢1 , 𝑘𝑢2
(a). Compute 𝐮 + 𝐯 and 𝑘𝐮 for 𝐮 = (1, −2), 𝐯 = (2, 0), and
𝑘 = 3.
𝐮 + 𝐯 = 1, −2 + 2, 0 = (2, −3)
𝑘𝐮 = 3 1, −2 = 3, −6
11
3.
Let V be the set of all ordered pairs of real numbers, and
consider the following addition and scalar multiplication
operations on 𝐮 = 𝑢1 , 𝑢2 and 𝐯 = 𝑣1 , 𝑣2 :
𝐮 + 𝐯 = 𝑢1 + 𝑣1 − 1, 𝑢2 + 𝑣2 − 1 , 𝑘𝐮 = 𝑘𝑢1 , 𝑘𝑢2
(b). Show that (0, 0) ≠ 𝟎.
讓 𝐯 = 0, 0
根據題目定義
𝐮 + 𝐯 = 𝑢1 + 0 − 1, 𝑢2 + 0 − 1 = (𝑢1 − 1, 𝑢2 − 1)
𝐮 + 0, 0 ≠ 𝐮 ，因此 (0, 0) ≠ 𝟎
12
3.
Let V be the set of all ordered pairs of real numbers, and
consider the following addition and scalar multiplication
operations on 𝐮 = 𝑢1 , 𝑢2 and 𝐯 = 𝑣1 , 𝑣2 :
𝐮 + 𝐯 = 𝑢1 + 𝑣1 − 1, 𝑢2 + 𝑣2 − 1 , 𝑘𝐮 = 𝑘𝑢1 , 𝑘𝑢2
(c). Show that 1, 1 = 𝟎.
讓 𝐯 = 1, 1
根據題目定義
𝐮 + 𝐯 = 𝑢1 + 1 − 1, 𝑢2 + 1 − 1 = (𝑢1 , 𝑢2 )
𝐮 + 1, 1 = 𝐮 ，因此 1, 1 = 𝟎
13
4.
Find the area of the triangle in 3-space that has the given
vertices: 𝑃 2, 6, −1 , 𝑄 1, 1, 1 , 𝑅 4, 6, 2 .
在 𝑅3 上
𝑃𝑄 = −1, −5, 2 ,
𝑃𝑅 = 2, 0, 3
𝑃𝑄 × 𝑃𝑅 = −1, −5, 2 × 2, 0, 3
−1 2 −1 −5
−5 2
=
,−
,
2 3
0 3
2
0
1
2
𝑃𝑄 × 𝑃𝑅 =
1
2
−15
2
+ −7
= −15, −7, 10
2
+ 102 =
374
2
14
5.
Describe the definition of a vector space using
no more than 30 words.
A nonempty set(4%) of objects equipped
with two operations, called addition(4%) and
scalar multiplication(4%) , satisfying the 10
axioms given in Section 4.1(4%)
15