Linear Algebra, 2020a

Section II. Linear Independence 117
Ŝ ⊂ S
Ŝ ⊃ S
S independent Ŝ must be independent Ŝ may be either
S dependent Ŝ may be either Ŝ must be dependent
Example 1.16 has something else to say about the interaction between linear
independence and superset. It names a linearly independent set that is maximal
in that it has no supersets that are linearly independent. By Lemma 1.15 a
linearly independent set is maximal if and only if it spans the entire space,
because that is when all the vectors in the space are already in the span. This
nicely complements Lemma 1.14, that a spanning set is minimal if and only if it
is linearly independent.
Exercises
̌ 1.21 Decide whether each subset of R 3 is linearly dependent or linearly independent.
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 2 4
1 2 3
0 1
(a) { ⎝−3⎠ , ⎝2⎠ , ⎝−4⎠} (b) { ⎝7⎠ , ⎝7⎠ , ⎝7⎠} (c) { ⎝ 0 ⎠ , ⎝0⎠}
5 4 14
7 7 7
−1 4
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
9 2 3 12
(d) { ⎝9⎠ , ⎝0⎠ , ⎝ 5 ⎠ , ⎝ 12 ⎠}
0 1 −4 −1
̌ 1.22 Which of these subsets of P 3 are linearly dependent and which are independent?
(a) {3 − x + 9x 2 ,5− 6x + 3x 2 ,1+ 1x − 5x 2 }
(b) {−x 2 ,1+ 4x 2 }
(c) {2 + x + 7x 2 ,3− x + 2x 2 ,4− 3x 2 }
(d) {8 + 3x + 3x 2 ,x+ 2x 2 ,2+ 2x + 2x 2 ,8− 2x + 5x 2 }
1.23 Determine if each set is linearly independent in the natural space.
⎛ ⎞ ⎛ ⎞
1 −1
(a) { ⎝2⎠ , ⎝ 1 ⎠} (b) {(1 3 1), (−1 4 3), (−1 11 7)}
0 0
( ) ( ) ( )
5 4 0 0 1 0
(c) { , , }
1 2 0 0 −1 4
̌ 1.24 Prove that each set {f, g} is linearly independent in the vector space of all
functions from R + to R.
(a) f(x) =x and g(x) =1/x
(b) f(x) =cos(x) and g(x) =sin(x)
(c) f(x) =e x and g(x) =ln(x)
̌ 1.25 Which of these subsets of the space of real-valued functions of one real variable
is linearly dependent and which is linearly independent? (We have abbreviated
some constant functions; e.g., in the first item, the ‘2’ stands for the constant
function f(x) =2.)