Abstract Linear Algebra, Fall 2011 - Sample Problems for Midterm I
1. Let V be a vector space and let v1 , . . . , vn be vectors in V .
a. What does it mean to say that v1 , . . . , vn are (i) linearly independent, (ii) generate (or
span) V , (iii) form a basis of V ?
b. Write ei for the vector in Rn with 1 in the i-th position and 0 in all other positions
(for i = 1, . . . , n). Show that e1 , . . . , en is a basis of Rn . Show also that
e1 − e2 , e2 − e3 , . . . , en−1 − en , e1 + e2 + · · · + en
n
is a basis of R .
2. a. What does it mean to say that a vector space is finite-dimensional?
b. Give an example with justification of a vector space that is not finite-dimensional.
c. In the zoo of all vector spaces lurks a strange space that does not have a basis. What
is this vector space?
3. Let t : V → W be a linear map.
a. What is the kernel ker t of t? Show that ker t is a subspace of V .
b. Let v, v 0 ∈ V . Show that t(v) = t(v 0 ) if and only if v = v 0 + u for some u ∈ ker t.
4. Let t : V → W be a linear map.
a. Let W1 be a subspace of W . Show that
t−1 (W1 ) = {v ∈ V | t(v) ∈ W1 }
is a subspace of V . Can you say why this problem generalizes (the second part of)
3 a?
b. Let V1 be a subspace of V and set
t(V1 ) = {t(v) | v ∈ V1 }.
(Thus w ∈ t(V1 ) means w = t(v) for some v ∈ V1 ). Show that t(V1 ) is a subspace of
W.
5. Let U and W be vector subspaces of a vector space V . Recall that V = U ⊕ W if for
each v ∈ V there exist unique vectors u ∈ U and w ∈ W such that v = u + w.
a. Show that V = U ⊕ W if and only if (i) V = U + W and (ii) U ∩ W = {0}.
b. Consider the subspace H of R4 given by
U = {(x1 , x2 , x3 , x4 ) ∈ R4 | x1 + x2 = 0, x3 + x4 = 0}.
Find a subspace W of R4 such that R4 = U ⊕ W .
6. Let V be real vector space (i.e., a vector space over R) of dimension n and let t : V → R
be a non-zero linear map (so that t(v) 6= 0 for some v ∈ V ).
a. Determine dim ker t.
2
b. Suppose t(v) 6= 0 for v ∈ V . Show that V = ker t ⊕ Rv where Rv = {αv | α ∈ R}.
c. Let s : V → R be a linear map such that ker s = ker t. Show that there is an α ∈ R
such that s = α t. (Hint: first show s(v) = αt(v), for some α ∈ R, and then use part
b).
7. Let t : R4 → R2 be a linear map such that
ker t = {(x1 , x2 , x3 , x4 ) | x1 = x2 , x3 = 2x4 }.
Is t surjective? Explain your reasoning.
8. Let A = (aij ) be an n × n matrix (over a field F ). Define the transpose A> of A to
be the n × n matrix (over F ) whose ij entry is aji . Thus A> is obtained from A by
interchanging rows and columns.
a. Show that A 7→ A> : Mn (F ) → Mn (F ) is a linear map.
b. Let A and B be n × n matrices over F . Show that (AB)> = B > A> .
9. a. Write J for the n × n matrix in which every entry is 1. Determine J 2 . Determine
J 100 .
b. Let A = (αij ) denote the n × n matrix with zeros on the diagonal and ones off the
diagonal, i.e.,
(
0, if i = j,
αij =
1 if i 6= j.
Show that A2 = (n − 2)J + I where I = (δij ) is the n × n identity matrix.
10. Let S denote the real vector space of all infinite sequences of real numbers. The elements
of S are infinite tuples (α0 , α1 , α2 , . . .) which we write as (αn )∞
n=0 or simply (αn ). Let U
denote the set of all sequences (αn ) in S such that αn = 2αn−1 + αn−2 for n ≥ 2.
a. Show that U is a subspace of S of dimension 2.
√
√
b. Let η1 = 1 + 2, η2 = 1 − 2. Show that w1 = (η1n ), w2 = (η2n ) is a basis of U.
c. Let (xn )∞
n=0 be the infinite sequence with x0 = 2, x1 = 2 and xn = 2xn−1 + xn−2 for
n ≥ 2. Show that
xn = η1n + η2n (n ≥ 0).