A First Course in Linear Algebra, 2017a

216 R n
Theorem 4.114:
Let A be an m × n matrix. The following are equivalent.
1. rank(A)=n.
2. row(A)=R n , i.e., the rows of A span R n .
3. The columns of A are independent in R m .
4. The n × n matrix A T A is invertible.
5. There exists an n × m matrix C so that CA = I n .
6. If A⃗x =⃗0 m for some⃗x ∈ R n ,then⃗x =⃗0 n .
Theorem 4.115:
Let A be an m × n matrix. The following are equivalent.
1. rank(A)=m.
2. col(A)=R m , i.e., the columns of A span R m .
3. The rows of A are independent in R n .
4. The m × m matrix AA T is invertible.
5. There exists an n × m matrix C so that AC = I m .
6. The system A⃗x =⃗b is consistent for every⃗b ∈ R m .
Exercises
Exercise 4.10.1 Here are some vectors.
⎡
⎣
⎤ ⎡
1
1 ⎦, ⎣
−2
1
2
−2
⎤
⎡
⎦, ⎣
2
7
−4
⎤
⎡
⎦, ⎣
5
7
−10
⎤
⎡
⎦, ⎣
12
17
−24
Describe the span of these vectors as the span of as few vectors as possible.
Exercise 4.10.2 Here are some vectors.
⎡
⎣
⎤ ⎡
1
2 ⎦, ⎣
−2
12
29
−24
⎤
⎡
⎦, ⎣
1
3
−2
⎤
⎡
⎦, ⎣
2
9
−4
⎤
⎡
⎦, ⎣
5
12
−10
Describe the span of these vectors as the span of as few vectors as possible.
⎤
⎦
⎤
⎦,