Linear Algebra, 2020a

134 Chapter Two. Vector Spaces
(c) a + b + c = 0, a + b − c = 0, and d ∈ R
̌ 2.20 Find the dimension of this subspace of R 2 .
( ) a + b
S = { | a, b, c ∈ R}
a + c
̌ 2.21 Find the dimension of each.
(a) The space of cubic polynomials p(x) such that p(7) =0
(b) The space of cubic polynomials p(x) such that p(7) =0 and p(5) =0
(c) The space of cubic polynomials p(x) such that p(7) =0, p(5) =0, and p(3) =0
(d) The space of cubic polynomials p(x) such that p(7) =0, p(5) =0, p(3) =0,
and p(1) =0
2.22 What is the dimension of the span of the set {cos 2 θ, sin 2 θ, cos 2θ, sin 2θ}? This
span is a subspace of the space of all real-valued functions of one real variable.
2.23 Find the dimension of C 47 , the vector space of 47-tuples of complex numbers.
2.24 What is the dimension of the vector space M 3×5 of 3×5 matrices?
̌ 2.25 Show that this is a basis for R 4 .
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 1 1 1
〈 ⎜0
⎟
⎝0⎠ , ⎜1
⎟
⎝0⎠ , ⎜1
⎟
⎝1⎠ , ⎜1
⎟
⎝1⎠ 〉
0 0 0 1
(We can use the results of this subsection to simplify this job.)
2.26 Decide if each is a basis for P 2 .
(a) {1, x 2 ,x 2 − x} (b) {x 2 + x, x 2 − x} (c) {2x 2 + x + 1, 2x + 1, 2}
(d) {3x 2 , −1, 3x, x 2 − x}
2.27 Refer to Example 2.11.
(a) Sketch a similar subspace diagram for P 2 .
(b) Sketch one for M 2×2 .
̌ 2.28 Where S is a set, the functions f: S → R form a vector space under the natural
operations: the sum f + g is the function given by f + g (s) =f(s)+g(s) and the
scalar product is r · f (s) =r · f(s). What is the dimension of the space resulting for
each domain?
(a) S = {1} (b) S = {1, 2} (c) S = {1,...,n}
2.29 (See Exercise 28.) Prove that this is an infinite-dimensional space: the set of
all functions f: R → R under the natural operations.
2.30 (See Exercise 28.) What is the dimension of the vector space of functions
f: S → R, under the natural operations, where the domain S is the empty set?
2.31 Show that any set of four vectors in R 2 is linearly dependent.
2.32 Show that 〈⃗α 1 , ⃗α 2 , ⃗α 3 〉⊂R 3 is a basis if and only if there is no plane through
the origin containing all three vectors.
2.33 Prove that any subspace of a finite dimensional space is finite dimensional.
2.34 Where is the finiteness of B used in Theorem 2.4?
2.35 Prove that if U and W are both three-dimensional subspaces of R 5 then U ∩ W
is non-trivial. Generalize.