# Linear Independence in vector space

**Definition.**

The vectors v

_{1}, v_{2}, ..., v_{3}in the vector**v**space are called linearly independent if
c

_{1}**v**_{1}+ c_{2}**v**_{2}+ ... + c_{n}**v**_{n}= 0
resulting in all scalars c

_{1}, c_{2}, ..., c_{3}must be equal to 0.**Example**

**1**

**.**

Then,

c

_{1}+ c_{2}= 0
c

_{1}+ 2c_{2}= 0
And the only solution to
this system is c

_{1}= 0, c_{2}= 0.**Definition.**

Vectors v

_{1}, v_{2}, ..., v_{n}in vector**v**space are called linearly dependent if there are scalars c_{1}, c_{2}, ..., c_{n}which are not all zero so
c

_{1}v_{1}+ c_{2}v_{2}+ ... + c_{n}v_{n}= 0**Example 2.**

Determine what vectors are

v

_{1}= (1, -2, 3) v_{2}= (5, 6, -1) v_{3}= (3, 2, 1)
linear Independent or linearly
dependent?

**Solution:**

On the vector component
segment

c

_{1}v_{1}+ c_{2}v_{2}+ c_{3}v_{3}= 0
to be

c

_{1}= (1, -2, 3) + c_{2}(5, 6, -1) + c_{3}(3, 2, 1) = (0, 0, 0)
or equivalent to

(c

_{1}+ 5c_{2}+ 3c_{3}, -2c_{1}+ 6c_{2}+ 2c_{3}, 3c_{1}– c_{2}+ c_{3}) = (0, 0, 0)
By equalizing the corresponding
component will give

c

_{1}+ 5c_{2}+ 3c_{3}= 0
2c

_{1}+ 6c_{2}+ 2c_{3}= 0
3c

_{1}– c_{2}+ c_{3}= 0
By completing this system
it will produce

c

_{1}= -1/2t c_{2}= -1/2t c_{3}= t
So, this system has
non-trivial solutions, so the three vectors are linearly dependent.

**Theorem 1.**

Suppose x

_{1}, x_{2}, ..., x_{n}are**n**vectors in R^{n}and for example
x

_{1}= (x_{1i}, x_{2i}, ..., x_{ni})^{T}untuk i = 1, 2, ..., n
if X = (x

_{1}, x_{2}, ..., x_{n}) then vectors x_{1}, x_{2}, ..., x_{n}are linearly dependent if and only if X is singular.**Proof:**

The equation c

_{1}x_{1}+ c_{2}x_{2}+ ... + c_{n}x_{n}= 0 is equivalent to the system of equations
If we say c = (c

_{1}, c_{2}, ..., c_{n})^{T}, then this system can be written as a matrix equation.
Xc = 0

This equation will have a
non-trivial solution if and only if

**X**is singular. So x_{1}, x_{2}, ..., x_{n}will depend linearly if and only if**is singular.***X***Example 3.**

Determine whether vectors (4,
2, 3)

^{T}, (2, 3, 1)^{T}and (2, -5. 3)^{T}are linearly dependent or not.

**Solution:**

Because

Then these vectors are
linear dependent.

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