MTH501 Linear Algebra Assignments No. 1 Solutions Fall 2014

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1 Linear combinations
Definition 1.1. Let V be a vector space. A vector v ∈ V is a linear combination of vectors
u1, u2, . . . , un if there exist a1, a2, . . . , an ∈ k such that
v = a1u1 + a2u2 + · · · + anun. (1)
Sometimes it is possible to express a vector as a linear combination of other vectors. It
can be done by solving a corresponding linear system. We’ll demonstrate it in the following
example.
Example 1.2. Consider the space R
2 — the space of all pairs of numbers. Let v = (8, 13),
u1 = (1, 2), and u2 = (2, 3). Let’s express v as a linear combination of u1 and u2. To do this we
have to find a and b such that v = au1+bu2, i.e. (8, 13) = a(1, 2)+b(2, 3) = (a·1+b·2, a·2+b·3).
So, we get the following system:
(
1a + 2b = 8
2a + 3b = 13
We can simply solve this system: subtracting the first equation multiplied by 2 from the second
one we get −b = −3, so b = 3, and so a = 8−2b = 2. So we see that (8, 13) = 2·(1, 2)+3·(2, 3).
Example 1.3. Consider the space P(t) — space of all polynomials. Let v = 5t
2 + 2t + 1,
u1 = t
2 + t, u2 = t + 1, u3 = t
2 + 1. Let’s express v as a linear combination of u1, u2
and u3. We should find a, b and c such that v = au1 + bu2 + cu3, i.e. 5t
2 + 2t + 1 =
a(t
2 + t) + b(t + 1) + c(t
2 + 1) = t
2
(a + c) + t(a + b) + (b + c). So, we get the following system:



a + c = 5
a + b = 2
b + c = 1
2 Linear dependence and independence
Now we’ll study one of the most important concepts of linear algebra and the theory of vector
spaces. This is a concept of linear dependence and independence.
Definition 2.1. Let u1, u2, . . . , un be a system of vectors. A linear combination of them is
called nontrivial if there exists a nonzero coefficient. If all coefficients are equal to 0, the
linear combination is called trivial.
Example 2.2. u1 + 0u2 + 0u3 −3u4 is nontrivial linear combination, and 0u1 + 0u2 + 0u3 + 0u4
is a trivial linear combination.
Definition 2.3. A system of vectors u1, u2, . . . , un is called linearly dependent if there exists
a nontrivial linear combination of these vectors which is equal to zero. Otherwise the system is
called linearly independent.
Example 2.4. Consider a vector space R
3
. Let u1 = (3, −5, 0), u2 = (5, 0, 1), and u3 =
(8, −5, 1). Then linear combination with coefficients 1,1, and -1 is nontrivial and equals to
zero:
1 · (3, 5, 0) + 1 · (5, 0, 1) + (−1) · (8, −5, 1) = (0, 0, 0).
Example 2.5. Consider a vector space R
2
. Let u1 = (1, 1), and u2 = (0, 0). The linear
combination with coefficients 0 and 1 is nontrivial, and equals to zero:
0 · (1, 1) + 1 · (0, 0) = (0, 0)
Moreover, if one of the vectors in the system equals to 0, then this system is linearly dependent,
since we can make a coefficient before it equal to some nonzero number, and all other coefficients
we can make equal to zero.