Alg1231W4T4 - Linear independence of orthogonal vectors

Friday, 17 August 2018

1:09 AM

Untitled picture.png Machine generated alternative text:
Recall that two vectors VI — 
IS zero. 
3 
and v 2 ¯ 
in the plane are orthogonal, or perpendicular, precisely when their dot product 
VI +YIY2 
Thus for example 
is orthogonal to [select all that apply] 
5 
-10 
6 
¯ 14 
23 
5 
-8 
35 
-21 
In general to get a simple vector orthogonal to v 
we could just choose the vector 
or any multiple of it. 
Using this definition, the only vector which is orthogonal to itself in IR2 is the vector 
1 
Recall: the Maple notation for the vector 
2 

Untitled picture.png Machine generated alternative text:
In three dimensions we define orthogonality also in terms of the dot product. If VI — 
and v'2 
then the dot product is 
VI + YlY2 + ZIZ2, 
and again VI is defined to be orthogonal, or perpendicular, to v2 precisely when 
Thus for example 
4 
is orthogonal to 
23 
Using this definition, the only vector which is orthogonal to itself in IR3 is the vector 
Recall: the Maple notation for the vector 2 is < 1 , 2 , 3 >.
Untitled picture.png Machine generated alternative text:
In three dimensions we define orthogonality also in terms of the dot product. If VI — 
and v'2 
then the dot product is 
VI + YlY2 + ZIZ2, 
and again VI is defined to be orthogonal, or perpendicular, to v2 precisely when 
Thus for example 
4 
is orthogonal to 
23 
Using this definition, the only vector which is orthogonal to itself in IR3 is the vector 
Recall: the Maple notation for the vector 2 is < 1 , 2 , 3 >. 

Untitled picture.png Machine generated alternative text:
A set of vectors S {VI v 2' 
vectors are orthogonal, that is 
in a vector space V is an orthogonal set if each v 
for 'i j. 
is a non-zero vector, and if any two vectors distinct 
Which of the following sets of vectors are orthogonal? [select all that apply] 
10 
¯ 14 

Untitled picture.png Machine generated alternative text:
A setof three vectors {VI, v2,v3} in a vector space is an orthogonal set if each is vector is a non-zero vector; and if every vector is orthogonal with the other two. 
This means explicitly that 
5 
-2 
So for example 
Note: 
3 
4 
1 
1 
—1 
1 
is an orthogonal set in R 
Can you find a set of three orthogonal vectors in R 
which includes the vector 
2 
-2 
Such a set is 
• express your set of vectors using the Maple notation 
• don't forget to include< 2, -4, -2 > in your set 
Hint: Once you have one vector orthogonal to 
2 
—4 
—2 
you can use the cross product to find a third. 

Untitled picture.png Machine generated alternative text:
A setof three vectors {VI, v2,v3} in a vector space is an orthogonal set if each is vector is a non-zero vector; and if every vector is orthogonal with the other two. 
This means explicitly that 
5 
-2 
So for example 
Note: 
3 
4 
1 
1 
—1 
1 
is an orthogonal set in R 
Can you find a set of three orthogonal vectors in R 
which includes the vector 
2 
-2 
Such a set is 
• express your set of vectors using the Maple notation 
• don't forget to include< 2, -4, -2 > in your set 
Hint: Once you have one vector orthogonal to 
2 
—4 
—2 
you can use the cross product to find a third. 

Untitled picture.png Machine generated alternative text:
We saw in the previous question that the set 
3 
4 
1 
5 
is an orthogonal set of vectors Let's illustrate the lesson of the video, that an orthogonal sets of vectors is 
Please complete the following argument 
We suppose first that we have a combination of the form 
for some constants r, s and t. To show that {v, u, w} is a linearly independent set; we want to show that 
To show that r = 0, we take the dot product of both sides of (1) with the vector v. This gives 
1 
—1 
1 
always linearly independent 
(1) 
we must nave 
(2) 
Now, looking at the vectors in S above, we see that 
26 
Substituting these values into equation (2) gives 
from which we easily deduce that r must equal (b 
Employing exactly the same reasoning, we deduce that we must have s 
So; since r = 0, s = = Ois tne only solution 
O and v w 
O and t 
we conclude that {v, 
IS a 
linearly independent set 
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Untitled picture.png Machine generated alternative text:
We saw in the previous question that the set 
3 
4 
1 
5 
is an orthogonal set of vectors Let's illustrate the lesson of the video, that an orthogonal sets of vectors is 
Please complete the following argument 
We suppose first that we have a combination of the form 
for some constants r, s and t. To show that {v, u, w} is a linearly independent set; we want to show that 
To show that r = 0, we take the dot product of both sides of (1) with the vector v. This gives 
1 
—1 
1 
always linearly independent 
(1) 
we must nave 
(2) 
Now, looking at the vectors in S above, we see that 
26 
Substituting these values into equation (2) gives 
from which we easily deduce that r must equal (b 
Employing exactly the same reasoning, we deduce that we must have s 
So; since r = 0, s = = Ois tne only solution 
O and v w 
O and t 
we conclude that {v, 
IS a 
linearly independent set 

Untitled picture.png Machine generated alternative text:
It is also possible to define functions as being orthogonal if we can find the correct definition to replace the dot product 
If we take two functions f and g , defined over the interval [0, 2%] , then we can define a fancy inner product 
This will play the same role as the dot product, taking tvo vectors as input and spitting out a number Using this; we define two functions to be orthogonal if 
The same way we do for geometric vectors. 
Let's consider the functions 
Using the inner product definition above we can determine 
sin(6x) = o 
f(x) 
= sin(6x), 
Showing that f@) 
sin(6c) and g@) 
sin(4x) are orthogonal with respect to this cool new product. 
Note: you may need to remember that the Maple notation for is Pi. 
Hint: 
• you might find it easier to evaluate the integrals above if you expand the product 
—i6x 
sin(4c). 
i4x 
2 
and (sin(4x))2 
• this trick also works for (sin(6x)) 
i6x 
sin(6x) sin(4x) 
and should yield familiar relations. 
—i4x 

Untitled picture.png Machine generated alternative text:
Use the hint provided, or if you're running out of time, you can simply use 
WolframAlpha! 
Also, to integrate or sin2(3x), you can convert it to: 
1 
2 
1 
2 
— cos(16x))
Untitled picture.png Machine generated alternative text:
Use the hint provided, or if you're running out of time, you can simply use 
WolframAlpha! 
Also, to integrate or sin2(3x), you can convert it to: 
1 
2 
1 
2 
— cos(16x))

 

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