Linear Algebra, 2020a

118 Chapter Two. Vector Spaces
(a) {2, 4 sin 2 (x), cos 2 (x)} (b) {1, sin(x), sin(2x)} (c) {x, cos(x)}
(d) {(1 + x) 2 ,x 2 + 2x, 3} (e) {cos(2x), sin 2 (x), cos 2 (x)} (f) {0, x, x 2 }
1.26 Does the equation sin 2 (x)/ cos 2 (x) =tan 2 (x) show that this set of functions
{sin 2 (x), cos 2 (x), tan 2 (x)} is a linearly dependent subset of the set of all real-valued
functions with domain the interval (−π/2..π/2) of real numbers between −π/2 and
π/2)?
1.27 Is the xy-plane subset of the vector space R 3 linearly independent?
̌ 1.28 Show that the nonzero rows of an echelon form matrix form a linearly independent
set.
1.29 (a) Show that if the set {⃗u,⃗v, ⃗w} is linearly independent then so is the set
{⃗u, ⃗u + ⃗v, ⃗u + ⃗v + ⃗w}.
(b) What is the relationship between the linear independence or dependence of
{⃗u,⃗v, ⃗w} and the independence or dependence of {⃗u − ⃗v,⃗v − ⃗w, ⃗w − ⃗u}?
1.30 Example 1.11 shows that the empty set is linearly independent.
(a) When is a one-element set linearly independent?
(b) How about a set with two elements?
1.31 In any vector space V, the empty set is linearly independent. What about all
of V?
1.32 Show that if {⃗x, ⃗y,⃗z} is linearly independent then so are all of its proper
subsets: {⃗x, ⃗y}, {⃗x,⃗z}, {⃗y,⃗z}, {⃗x},{⃗y}, {⃗z}, and {}. Is that ‘only if’ also?
1.33 (a) Show that this
⎛ ⎞ ⎛ ⎞
1 −1
S = { ⎝ ⎠ , ⎝
is a linearly independent subset of R 3 .
(b) Show that
⎛ ⎞
3
⎝2⎠
0
is in the span of S by finding c 1 and c 2 giving a linear relationship.
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 −1 3
c 1
⎝1⎠ + c 2
⎝ 2 ⎠ = ⎝2⎠
0 0 0
1
0
Show that the pair c 1 ,c 2 is unique.
(c) Assume that S is a subset of a vector space and that ⃗v is in [S], so that ⃗v is a
linear combination of vectors from S. Prove that if S is linearly independent then
a linear combination of vectors from S adding to ⃗v is unique (that is, unique up
to reordering and adding or taking away terms of the form 0 · ⃗s). Thus S as a
spanning set is minimal in this strong sense: each vector in [S] is a combination
of elements of S a minimum number of times — only once.
(d) Prove that it can happen when S is not linearly independent that distinct
linear combinations sum to the same vector.
1.34 Prove that a polynomial gives rise to the zero function if and only if it is
the zero polynomial. (Comment. This question is not a Linear Algebra matter
2
0
⎠}