Linear Algebra

Section III. Basis and Dimension 123
2.17 Find the dimension of the vector space of matrices
� �
a b
c d
subject to each condition.
(a) a, b, c, d ∈ R
(b) a − b +2c =0andd ∈ R
(c) a + b + c =0,a + b − c =0,andd ∈ R
� 2.18 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
2.19 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.
2.20 Find the dimension of C 47 , the vector space of 47-tuples of complex numbers.
2.21 What is the dimension of the vector space M3×5 of 3×5 matrices?
� 2.22 Show that this is a basis for R 4 .
⎛ ⎞ ⎛ ⎞
1 1
⎛ ⎞
1
⎛ ⎞
1
⎜0⎟
⎜1⎟
⎜1⎟
⎜1⎟
〈 ⎝
0
⎠ , ⎝
0
⎠ , ⎝
1
⎠ , ⎝
1
⎠〉
0 0 0 1
(The results of this subsection can be used to simplify this job.)
2.23 Refer to Example 2.9.
(a) Sketch a similar subspace diagram for P2.
(b) Sketch one for M2×2.
� 2.24 Observe that, where S is a set, the functions f : S → R form a vector space
under the natural operations: f + g (s) =f(s)+g(s) andr · 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}
2.25 (See Exercise 24.) Prove that this is an infinite-dimensional space: the set of
all functions f : R → R under the natural operations.
2.26 (See Exercise 24.) 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?
2.27 Show that any set of four vectors in R 2 is linearly dependent.
2.28 Show that the set 〈�α1,�α2,�α3〉 ⊂R 3 is a basis if and only if there is no plane
through the origin containing all three vectors.
2.29 (a) Prove that any subspace of a finite dimensional space has a basis.
(b) Prove that any subspace of a finite dimensional space is finite dimensional.
2.30 Where is the finiteness of B used in Theorem 2.3?
� 2.31 Prove that if U and W are both three-dimensional subspaces of R 5 then U ∩W
is non-trivial. Generalize.
2.32 Because a basis for a space is a subset of that space, we are naturally led to
how the property ‘is a basis’ interacts with set operations.
(a) Consider first how bases might be related by ‘subset’. Assume that U, W are