vectors in ndimensional space

8/14/2019 Vectors in Ndimensional Space

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http://programeansic.blogspot.comVECTORS IN

In the first two sections of this chapter we looked at vectors in 2-space and 3-space. You

probably noticed that with the exception of the cross product (which is only defined in 3-space)

all of the formulas that we had for vectors in 3-space were natural extensions of the 2-spaceformulas. In this section were going to extend things out to a much more general setting. We

wont be able to visualize things in a geometric setting as we did in the previous two sections butthings will extend out nicely. In fact, that was why we started in 2-space and 3-space. We

wanted to start out in a setting where we could visualize some of what was going on before we

generalized things into a setting where visualization was a very difficult thing to do.

So, lets get things started off with the following definition.

Definition 1 Given a positive integern an ordered n-tuple is a sequence ofn real numbers

denoted by . The complete set of all ordered n-tuples is called n-space andis denoted by

n the previous sections we were looking at (what we were calling 2-space) and(what we were calling 3-space). Also the more standard terms for 2-tuples and 3-tuples

are ordered pair and ordered triplet and thats the terms well be using from this point on.

Also, as we pointed out in the previous sections an ordered pair, , or an ordered

triplet, , can be thought of as either a point or a vector in or respectively.

In general an ordered n-tuple, , can also be thought of as a point or a

vector in . Again, we cant really visualize a point or a vector in , but we will think of

them as points or vectors in anyway and try not to worry too much about the fact that we

cant really visualize them.

Next, we need to get the standard arithmetic definitions out of the way and all of these are going

to be natural extensions of the arithmetic we saw in and .http://programeansic.blogspot.com
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Definition 2 Suppose and

are two vectors in .(a) We say that u and v are equal if,

(b) The sum ofu and v is defined to be,

(c) The negative (or additive inverse) ofu is defined to be,

(d) The difference of two vectors is defined to be,

(e) Ifc is any scalar then the scalar multiple ofu is defined to be,

(f) The zero vector in is denoted by 0 and is defined to be,

The basic properties of arithmetic are still valid in so lets also give those so that we can say

that weve done that.

Theorem 1 Suppose ,

and are vectors in and c and kare scalars then,(a

(b

(d \\

(e)

(f)

(g

(h

.

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We now need to extend the dot product we saw in the previous section to and well be

giving it a new name as well.


are two vectors in then the Euclidean inner product denoted by is defined to be

So, we can see that its the same notation and is a natural extension to the dot product that welooked at in the previous section, were just going to call it something different now. In fact, this

is probably the more correct name for it and we should instead say that weve renamed this to the

dot product when we were working exclusively in and .

Note that when we add in addition, scalar multiplication and the Euclidean inner product to

we will often call this Euclidean n-space.We also have natural extensions of the properties of the dot product that we saw in the previous

section.

Theorem 2 Suppose , , and

are vectors in and let c be a scalar then,

(a)(b)

(c)(d)

(c)

(e) and(f)

(e) if and only ifu=0.

The final extension to the work of the previous sections that we need to do is to give the

definition of the norm for vectors in and well use this to define distance in .

Definition 4 Suppose is a vector in then the Euclidean norm is,



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are two points in then the Euclidean distancebetween them is defined to be,

Notice in this definition that we called u and v points and then used them as vectors in the norm.

This comes back to the idea that an n-tuple can be thought of as both a point and a vector and sowill often be used interchangeably where needed.

Lets take a quick look at a couple of examples.

Example 1 Given and

compute

(a)

(b)

(c)

(d)

(e)

Solution

There really isnt much to do here other than use the appropriate definition.(a)

(b)

(c)



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(d)

(e)

Definition of unit vector :-Just as we saw in the section onvectorsif we have then we

will call u a unit vector

And so the vectoru from the previous set of examples is not a unit vector

Now that weve gotten both the inner product and the norm taken care of we can give the

following theorem.

Theorem 3 Suppose u and v are two vectors in and is the angle between them. Then,

Of course since we are in it is hard to visualize just what the angle between the two vectors

is, but provided we can find it we can use this theorem. Also note that this was the definition of

the dot product that we gave in the previous section and like that section this theorem is mostuseful for actually determining the angle between two vectors.

The next theorem is very important and has many uses in the study of vectors. In fact well need

it in the proof of at least one theorem in these notes. The following theorem is called

the Cauchy-Schwarz Inequality.

Theorem 4 Suppose u and v are two vectors in then

Proof : This proof is surprisingly simple. Well start with the result of the previous theorem and

take the absolve value of both sides.

http://tutorial.math.lamar.edu/Classes/LinAlg/Vectors.aspx#UnitVectorhttp://tutorial.math.lamar.edu/Classes/LinAlg/Vectors.aspx#UnitVectorhttp://tutorial.math.lamar.edu/Classes/LinAlg/Vectors.aspx#UnitVectorhttp://tutorial.math.lamar.edu/Classes/LinAlg/Vectors.aspx#UnitVector


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However, we know that

and so we get our result by using this fact.

Here are some nice properties of the Euclidean norm.

Theorem 5 Suppose u and v are two vectors in and that c is a scalar then,

(a

(b if and only ifu=0.

(c)

(d - Usually called the Triangle Inequality

The proof of the first two part is a direct consequence of the definition of the Euclidean norm andso wont be given here.

Proof :

(c) Well just run through the definition of the norm on this one.

(d) The proof of this one isnt too bad once you see the steps you need to take. Well start withthe following.



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So, were starting with the definition of the norm and squaring both sides to get rid of the square

root on the right side. Next, well use the properties of the Euclidean inner product to simplify

this.

Now, notice that we can convert the first and third terms into norms so well do that. Also,

is a number and so we know that if we take the absolute value of this well

have . Using this and converting the first and third terms to norms gives,

We can now use the Cauchy-Schwarz inequality on the second term to get,

Were almost done. Lets notice that the left side can now be rewritten as,

Finally, take the square root of both sides.

Example 2 Given and verify the Cauchy-

Schwarz inequality and the Triangle Inequality.



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Solution

Lets first verify the Cauchy-Schwarz inequality. To do this we need to following quantities.

Now, verify the Cauchy-Schwarz inequality.

Sure enough the Cauchy-Schwarz inequality holds true.

To verify the Triangle inequality all we need is,

Now verify the Triangle Inequality.

So, the Triangle Inequality is also verified for this problem.

Here are some nice properties pertaining to the Euclidean distance.

Theorem 6 Suppose u, v, and w are vectors in then,

(a

(b if and only ifu=v.

(c)

(d - Usually called the Triangle Inequality

The proof of the first two parts is a direct consequence of the previous theorem and the proof ofthe third part is a direct consequence of the definition of distance and wont be proven here.

Proof ofTRIANGLE INEQUALITY ---http://programeansic.blogspot.com


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Lets start off with the definition of distance.

Now, add in and subtract out w as follows,

Next use the Triangle Inequality for norms on this.

Finally, just reuse the definition of distance again.

HENCE THE PROOF.

Now, We have one final topic that needs to be generalized into Euclidean n-space.

Definition 6 Suppose u and v are two vectors in . We say

that u and v are orthogonal if .

So, this definition of orthogonality is identical to the definition that we saw when we were

dealing with and .

Here is the Pythagorean Theorem in .

Theorem 7 Suppose u and v are two orthogonal vectors in then,

Proof : The proof of this theorem is fairly simple. From the proof of the triangle inequality for

norms we have the following statement.

However, because u and v are orthogonal we have and so we get,

he Hence the proof.



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Example 3 Show that and

are orthogonal and verify that the Pythagorean Theorem holds.

Solution

Showing that these two vectors is easy enough.

So, the Pythagorean Theorem should hold, but lets verify that. Heres the sum

---and heres the square of the norms.

A quick computation then confirms that .

Weve got one more theorem that gives a relationship between the Euclidean inner product and

the norm. This may seem like a silly theorem, but well actually need this theorem towards the

end of the next chapter.

Theorem 8 Ifu and v are two vectors in then,

Proof : The proof here is surprisingly simple. First, start with,

The first of these weve seen a couple of times already and the second is derived in the same

manner that the first was and so you should verify that formula.



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Now subtract the second from the first to get,

Finally, divide by 4 and we get the result we were after.

Hence the proof.

In the previous section we saw the three standard basis vectorsfor are i,j, and k. This

idea can also be extended out to . In we will define the standard basisvectors orstandard unit vectors to be,

and just as we saw in that section we can write any vector

in terms of these standard basis vectors as follows,

Note that in we have , and .

Now that weve gotten the general vector in Euclidean n-space taken care of we need to go back

and remember some of the work that we did in the first chapter. It is often convenient to write

the vector as either a row matrix or a column matrix as follows,

http://tutorial.math.lamar.edu/Classes/LinAlg/Dot_CrossProduct.aspx#StandardBasishttp://tutorial.math.lamar.edu/Classes/LinAlg/Dot_CrossProduct.aspx#StandardBasishttp://tutorial.math.lamar.edu/Classes/LinAlg/Dot_CrossProduct.aspx#StandardBasis


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In this notation we can use matrix addition and scalar multiplication for matrices to show thatwell get the same results as if wed done vector addition and scalar multiplication for vectors on

the original vectors.

So, why do we do this? Well lets use the column matrix notation for the two vectors u and v.

Now compute the following matrix product.

So, we can think of the Euclidean inner product can be thought of as a matrix multiplicationusing,

provided we consideru and v as column vectors.

The natural question this is just why is this important? Well lets consider the following

scenario. Suppose that u and vare two vectors in and that A is an matrix. Nowconsider the following inner product and write it as a matrix multiplication.

Now, rearrange the order of the multiplication and recall one of theproperties of transposes.

http://tutorial.math.lamar.edu/Classes/LinAlg/PropsOfMatrixArith.aspx#Sys_PropMat_Prop4http://tutorial.math.lamar.edu/Classes/LinAlg/PropsOfMatrixArith.aspx#Sys_PropMat_Prop4http://tutorial.math.lamar.edu/Classes/LinAlg/PropsOfMatrixArith.aspx#Sys_PropMat_Prop4


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Dont forget that we switch the order on the matrices when we move the transpose out of the

parenthesis. Finally, this last matrix product can be rewritten as an inner product.

This tells us that if weve got an inner product and the first vector (or column matrix) is

multiplied by a matrix then we can move that matrix to the second vector (or column matrix) if

we simply take its transpose. A similar argument can also show that,


So ,here its an end of this topic .now we look At linear transformation and vector space.

vectors in ndimensional space

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