Guided Proof Prove that a nonempty subset of a finite set of linearly independent vectors is linearly independent. Getting Started: You need to show that a subset of a linearly independent set of vectors cannot be linearly dependent. (i) Assume S is a set of linearly independent vectors. Let T be a subset of S . (ii) If T is linearly dependent, then there exist constants not all zero satisfying the vector equation c 1 v 1 + c 2 v 2 + ... + c k v k = 0 . (iii) Use this fact to derive a contradiction and conclude that T is linearly independent.
Guided Proof Prove that a nonempty subset of a finite set of linearly independent vectors is linearly independent. Getting Started: You need to show that a subset of a linearly independent set of vectors cannot be linearly dependent. (i) Assume S is a set of linearly independent vectors. Let T be a subset of S . (ii) If T is linearly dependent, then there exist constants not all zero satisfying the vector equation c 1 v 1 + c 2 v 2 + ... + c k v k = 0 . (iii) Use this fact to derive a contradiction and conclude that T is linearly independent.
Solution Summary: The author explains the proof of nonempty subsets of a finite set of linearly independent vectors.
Guided Proof Prove that a nonempty subset of a finite set of linearly independent vectors is linearly independent.
Getting Started: You need to show that a subset of a linearly independent set of vectors cannot be linearly dependent.
(i) Assume
S
is a set of linearly independent vectors. Let
T
be a subset of
S
.
(ii) If
T
is linearly dependent, then there exist constants not all zero satisfying the vector equation
c
1
v
1
+
c
2
v
2
+
...
+
c
k
v
k
=
0
.
(iii) Use this fact to derive a contradiction and conclude that
T
is linearly independent.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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