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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