Alg1231W4T1 - Linear Independence

Tuesday, 14 August 2018

8:02 PM

Untitled picture.png Machine generated alternative text:
A set of vectors {VI, , vn} is defined to be linearly independent if the QDJy. solution to
(1)
r IVI + r 2V2 = O
is given by
—o.
(2)
It's worth noting that (2) is always a solution to equation (1), so in fact all we need to do is prove that (1) has a unique solution. This gives us a clear
method for determining linear independence.
Let's determine whether the set
solutions of the equation
1
2
3
4
5
6
7
8
9
is a linearly independent set. From the discussion above, we know we need to consider the
1
3
4
6
7
9
As we all remember from Mathematics IA, we can find the general solution by considering the augmented matrix [select all that apply]
1
2
3
3r1
1
4
7
4
5
6
2
5
8
9r3
3
6
9

Untitled picture.png Machine generated alternative text:
If we row reduce this we get
1
o
o
4
—3
o
7
—6
o
o
We already know this system has some solutions (ruling out the 'no solution' option), and since
this system of equations actually has infinitely many solutions
Thus:
r.3 = 0, and so the set is linearly independent
C) the only solution is rl = r 2
@ there will be non-zero solutions, and so the set is not linearly independent.
O we can conclude that
the third column is non-leading
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Untitled picture.png Machine generated alternative text:
If we row reduce this we get
1
o
o
4
—3
o
7
—6
o
o
We already know this system has some solutions (ruling out the 'no solution' option), and since
this system of equations actually has infinitely many solutions
Thus:
r.3 = 0, and so the set is linearly independent
C) the only solution is rl = r 2
@ there will be non-zero solutions, and so the set is not linearly independent.
O we can conclude that
the third column is non-leading

Untitled picture.png Machine generated alternative text:
Kendra wants to figure out if the set S
then row reduces A to get U.
3
{VI , v2, v3} is linearly independent. Naturally she writes the three vectors as the columns of a matrix A , and
-I
-2
8
3
5
5
, then Kendra deduces that S is a linearly dependent
O set. This is because she sees that the equation
OVI + ßV2 + = O
has a non-zero
ii) If U
has only the zero
iii) If U
has a non-zero
0 solution, for example
[-20, 1, 6]
, then Kendra deduces that S is a linearly independent
0 set. This is because she sees that the equation
avi + ßV2 + IV3 = O
0 solution given by [0' P'
, then Kendra deduces that S is a linearly dependent
0 set. This is because she sees that the equation
avi + ßV2 + IV3 = O
0 solution, for example

Untitled picture.png Machine generated alternative text:
i) Is the set of vectors
linearly independent? If not, enter a non-zero solution to the equation
Otherwise enter the zero solution (i.e. [0 , 0] ).
[0,0]
ii) Is the set of vectors
Otherwise enter the zero solution (i.e. [0 , o , 0] ).
linearly independent? If not, enter a non-zero solution to the equation
1
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Untitled picture.png Machine generated alternative text:
i) Is the set of vectors
linearly independent? If not, enter a non-zero solution to the equation
Otherwise enter the zero solution (i.e. [0 , 0] ).
[0,0]
ii) Is the set of vectors
Otherwise enter the zero solution (i.e. [0 , o , 0] ).
linearly independent? If not, enter a non-zero solution to the equation
1

Untitled picture.png Machine generated alternative text:
iv) Is the set of vectors
-10
linearly independent? If not, enter a non-zero solution to the equation
5
5
—10
Otherwise enter the zero solution (i.e. [0 , o , 0] ).

Untitled picture.png Machine generated alternative text:
The vectors VI
6
-12
-9
-6
-15
-6
-12
12
4
ink are not linearly independent (i.e. they are linearly dependent). Use row reduction to find
-2
12
constants , not all zero, so that CIVI + 02 v '2 + c3v3 = O
An example of such is
We can convert such a linear relation into forms that express one of the vectors in terms of the others. Examples of such relationships between the vectors
are:
• VI ¯

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Untitled picture.png Machine generated alternative text:
The vectors VI
6
-12
-9
-6
-15
-6
-12
12
4
ink are not linearly independent (i.e. they are linearly dependent). Use row reduction to find
-2
12
constants , not all zero, so that CIVI + 02 v '2 + c3v3 = O
An example of such is
We can convert such a linear relation into forms that express one of the vectors in terms of the others. Examples of such relationships between the vectors
are:
• VI ¯

Untitled picture.png Machine generated alternative text:
In the RGB colour scheme based on additive colours Red, Green and Blue, we define
Other colours are then linear combinations of these, for example:
B = (0, o, 255).
C (0, 255,255) (Cyan)
M (Magenta)
• Y (Yellow).
In 3-dimensional colour space, {R, C, B} is a linearly
Which of the following are linearly independent sets?
{Lavender(230, 230, 250), M , R}
{C, R, Dark slate grey(47, 79, 79)}
O {White(255, 255, 255), O,
independent
O set, and {C, M, Y} is a linearly independent
O set.

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Untitled picture.png Machine generated alternative text:
In the RGB colour scheme based on additive colours Red, Green and Blue, we define
Other colours are then linear combinations of these, for example:
B = (0, o, 255).
C (0, 255,255) (Cyan)
M (Magenta)
• Y (Yellow).
In 3-dimensional colour space, {R, C, B} is a linearly
Which of the following are linearly independent sets?
{Lavender(230, 230, 250), M , R}
{C, R, Dark slate grey(47, 79, 79)}
O {White(255, 255, 255), O,
independent
O set, and {C, M, Y} is a linearly independent
O set.

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