1.7 Linear Independence

in Rn is said to be linearly independent if

has only the trivial solution.

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x1v1 + x2v2 +L + x pvp = 0

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v1,v2,L ,vp{ }

in Rn is said to be linearly dependent if

there exist weights , not all zero, such thatpccc ,,, 21

Definitions

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v1,v2,L ,vp{ }

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c1v1 + c2v2 +L + c pvp = 0

Examples: Determine if each of the following sets of vectors are linearly independent.

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1. v1 =12 ⎡ ⎣ ⎢

⎤ ⎦ ⎥ v2 =

−10

⎡ ⎣ ⎢

⎤ ⎦ ⎥

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2. v1 =10 ⎡ ⎣ ⎢

⎤ ⎦ ⎥ v2 =

−20

⎡ ⎣ ⎢

⎤ ⎦ ⎥

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3. v1 =12 ⎡ ⎣ ⎢

⎤ ⎦ ⎥ v2 =

−10

⎡ ⎣ ⎢

⎤ ⎦ ⎥ v3 =

01 ⎡ ⎣ ⎢

⎤ ⎦ ⎥

The columns of a matrix A are linearly independent if and only if

has only the trivial solution.

Examples: Determine if the columns of the following matrices are linearly independent.

4221

.2

4321

.1

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Ax = 0

Tips to determine the linear dependence

1. The set has two vectors and one is a multiple of the other.

2. The set has two or more vectors and one of the vectors is a linear combination of the others.

3. The set contains more vectors than the number of entries in each vector.

4. The set contains the zero vector.

A set of vectors are linearly dependent if any of the following are true: