4. State the converse to Theorem 8 (1.7 pg 63). Then, show the converse is false by providing a counterexample. Note: For the implication P→Q, the converse is Q→ P
4. State the converse to Theorem 8 (1.7 pg 63). Then, show the converse is false by providing a counterexample. Note: For the implication P→Q, the converse is Q→ P
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![n
THEOREM 8
FIGURE 3
If p > n, the columns are linearly
dependent.
(-2,2)
(2, 1)
(4,-1)
FIGURE 4
A linearly dependent set in R².
THEOREM 9
Example 4 generalizes to any set (u, v, W} in K" with u and V linearly independent.
The set {u, v, w} will be linearly dependent if and only if w is in the plane spanned by
u and v.
The next two theorems describe special cases in which the linear dependence of a
set is automatic. Moreover, Theorem 8 will be a key result for work in later chapters.
If a set contains more vectors than there are entries in each vector, then the set
is linearly dependent. That is, any set (v₁.....V₂) in R" is linearly dependent if
p>n.
PROOF Let A = [V₁ V₂]. Then A is n xp, and the equation Ax = 0 corre-
sponds to a system of n equations in p unknowns. If p > n, there are more variables
than equations, so there must be a free variable. Hence Ax = 0 has a nontrivial solution,
and the columns of A are linearly dependent. See Figure 3 for a matrix version of this
theorem.
Warning: Theorem 8 says nothing about the case in which the number of vectors in
the set does not exceed the number of entries in each vector.
EXAMPLE 5 The vectors
rs [²]-[-]-[-2] are linearly dependent by Theorem
8, because there are three vectors in the set and there are only two entries in each vector.
Notice, however, that none of the vectors is a multiple of one of the other vectors. See
Figure 4.
If a set 5 = {V₁..... Vp} in R" contains the zero vector, then the set is linearly
dependent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F21e98448-8eaa-4c09-a4ab-b543c4300c00%2Fea0b29d2-eeeb-4432-bea5-02ce24ddfd28%2F1prywgw_processed.png&w=3840&q=75)
Transcribed Image Text:n
THEOREM 8
FIGURE 3
If p > n, the columns are linearly
dependent.
(-2,2)
(2, 1)
(4,-1)
FIGURE 4
A linearly dependent set in R².
THEOREM 9
Example 4 generalizes to any set (u, v, W} in K" with u and V linearly independent.
The set {u, v, w} will be linearly dependent if and only if w is in the plane spanned by
u and v.
The next two theorems describe special cases in which the linear dependence of a
set is automatic. Moreover, Theorem 8 will be a key result for work in later chapters.
If a set contains more vectors than there are entries in each vector, then the set
is linearly dependent. That is, any set (v₁.....V₂) in R" is linearly dependent if
p>n.
PROOF Let A = [V₁ V₂]. Then A is n xp, and the equation Ax = 0 corre-
sponds to a system of n equations in p unknowns. If p > n, there are more variables
than equations, so there must be a free variable. Hence Ax = 0 has a nontrivial solution,
and the columns of A are linearly dependent. See Figure 3 for a matrix version of this
theorem.
Warning: Theorem 8 says nothing about the case in which the number of vectors in
the set does not exceed the number of entries in each vector.
EXAMPLE 5 The vectors
rs [²]-[-]-[-2] are linearly dependent by Theorem
8, because there are three vectors in the set and there are only two entries in each vector.
Notice, however, that none of the vectors is a multiple of one of the other vectors. See
Figure 4.
If a set 5 = {V₁..... Vp} in R" contains the zero vector, then the set is linearly
dependent.
Transcribed Image Text:4. State the converse to Theorem 8 (1.7 pg 63). Then, show the converse is false by providing a counterexample. Note:
For the implication P→Q, the converse is Q→→ P
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