nsider the set of vectors vị = 2 V2 = 1 and 4 e vectors vị and v2 are O Linearly Independent O Linearly Dependent e vectors vị and v3 are Linearly Independent O Linearly Dependent e vectors v2 and v3 are O Linearly Dependent

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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Part I
Consider the set of vectors vj =
1
2
V2 =
1
and
4
V3
1
|
The vectors vị and v2 are
Linearly Independent
Linearly Dependent
The vectors vị and vz are
Linearly Independent
Linearly Dependent
The vectorS V2 and vz are
Linearly Dependent
Linearly Independent
The vectors v1,v2,V3 are
Linearly Independent
Linearly Dependent
We can solve the equation c¡v1 + c2v2 + C3V3 = 0 using
coefficients that are not all zeros. For example, we could use c1 =
1,
C2
and c3 =
In general, if v1,V2 and v3 are the columns of a matrix A, then the set
of vectors v1, v2, V3 is linearly independent when the only solution of
Ax = 0 is the matrix x =
Even more generally, if v1, V2, ..., Vn are the columns of a matrix A,
then the set of vectors v1, V2, ..., Vn is linearly independent when the
matrix A has rank
Transcribed Image Text:Part I Consider the set of vectors vj = 1 2 V2 = 1 and 4 V3 1 | The vectors vị and v2 are Linearly Independent Linearly Dependent The vectors vị and vz are Linearly Independent Linearly Dependent The vectorS V2 and vz are Linearly Dependent Linearly Independent The vectors v1,v2,V3 are Linearly Independent Linearly Dependent We can solve the equation c¡v1 + c2v2 + C3V3 = 0 using coefficients that are not all zeros. For example, we could use c1 = 1, C2 and c3 = In general, if v1,V2 and v3 are the columns of a matrix A, then the set of vectors v1, v2, V3 is linearly independent when the only solution of Ax = 0 is the matrix x = Even more generally, if v1, V2, ..., Vn are the columns of a matrix A, then the set of vectors v1, V2, ..., Vn is linearly independent when the matrix A has rank
Part III
1 1 2
Consider the matrix A
2 2 1. The column space of A has
3 3 3
dimension
The dimension of the nullspace of A is
Transcribed Image Text:Part III 1 1 2 Consider the matrix A 2 2 1. The column space of A has 3 3 3 dimension The dimension of the nullspace of A is
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