A First Course in Linear Algebra, 2017a

4.10. Spanning, Linear Independence and Basis in R n 223
Exercise 4.10.31 Here are some vectors in R 4 .
⎡ ⎤ ⎡ ⎤ ⎡
1 3
⎢ b + 1
⎥
⎣ a ⎦ , ⎢ 3b + 3
⎥
⎣ 3a ⎦ , ⎢
⎣
1 3
1
b + 2
2a + 1
1
⎤ ⎡
⎥
⎦ , ⎢
⎣
2
2b − 5
−5a − 7
2
⎤ ⎡
⎥
⎦ , ⎢
⎣
1
b + 2
2a + 2
1
Thse vectors can’t possibly be linearly independent. Tell why. Next obtain a linearly independent subset of
these vectors which has the same span as these vectors. In other words, find a basis for the span of these
vectors.
⎧⎡
⎪⎨
Exercise 4.10.32 Let H = span ⎢
⎣
⎪ ⎩
2
1
1
1
⎤
⎡
⎥
⎦ , ⎢
⎤⎫
⎪⎬
⎥
⎦ . Find the dimension of H and de-
⎪⎭
termine a basis.
⎧⎡
⎪⎨
Exercise 4.10.33 Let H denote span ⎢
⎣
⎪⎩
and determine a basis.
⎧⎡
⎪⎨
Exercise 4.10.34 Let H denote span ⎢
⎣
⎪⎩
H and determine a basis.
⎧⎡
⎪⎨
Exercise 4.10.35 Let H denote span ⎢
⎣
⎪⎩
mension of H and determine a basis.
⎧⎡
⎪⎨
Exercise 4.10.36 Let H denote span ⎢
⎣
⎪⎩
H and determine a basis.
⎧⎡
⎪⎨
Exercise 4.10.37 Let H denote span ⎢
⎣
⎪⎩
and determine a basis.
⎣
0
1
1
−1
−2
1
1
−3
2
3
2
1
−1
0
−1
−1
⎤
⎤ ⎡
⎥
⎦ , ⎢
⎣
−1
1
−1
−2
⎤
⎤ ⎡
⎥
⎦ , ⎢
⎣
⎡
⎥
⎦ , ⎢
⎤
⎡
⎥
⎦ , ⎢
0
2
0
−1
⎡
⎥
⎦ , ⎢
⎣
⎣
−1
−1
−2
2
−9
4
3
−9
⎣
8
15
6
3
⎤ ⎡
⎥
⎦ , ⎢
⎣
5
2
3
3
⎤
⎡
⎥
⎦ , ⎢
⎤
−4
3
−2
−4
⎡
⎥ ⎥⎦ , ⎢
⎣
⎣
⎤ ⎡
⎥
⎦ , ⎢
⎣
⎤ ⎡
⎥
⎦ , ⎢
⎣
−1
6
0
−2
⎤
⎡
⎥
⎦ , ⎢
3
6
2
1
⎤ ⎡
⎥
⎦ , ⎢
⎣
−1
1
−2
−2
2
3
5
−5
−33
15
12
−36
⎣
⎤
⎥
⎦
⎤ ⎡
⎥
⎦ , ⎢
⎣
−3
2
−1
−2
⎤ ⎡
⎥
⎦ , ⎢
⎣
−2
16
0
−6
⎤ ⎡
⎥
⎦ , ⎢
⎣
⎤
⎡
⎥
⎦ , ⎢
4
6
6
3
⎤ ⎡
⎥
⎦ , ⎢
⎣
0
1
2
−2
⎣
⎤ ⎡
⎥
⎦ , ⎢
⎣
−22
10
8
−24
−1
1
−2
−4
−3
22
0
−8
⎤⎫
⎪⎬
⎥
⎦ . Find the dimension of H
⎪⎭
8
15
6
3
⎤⎫
⎪⎬
⎥
⎦ . Find the dimension of
⎪⎭
⎤
⎡
⎥
⎦ , ⎢
⎣
−7
5
−3
−6
⎤⎫
⎪ ⎬
⎥
⎦ . Find the di-
⎪ ⎭
⎤⎫
⎪⎬
⎥
⎦ . Find the dimension of
⎪⎭
⎤⎫
⎪⎬
⎥
⎦ . Find the dimension of H
⎪⎭