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| − | = '''<br>''' = | + | = '''<br>''' = |
| − | = '''Statement: I am going to show that if V is a subspace of R<sup>n</sup>. then dim(V)+dim(V<sup>orth</sup>)=n<br>''' = | + | = '''Statement: I am going to show that if V is a subspace of R<sup>n</sup>. then dim(V)+dim(V<sup>orth</sup>)=n<br>''' = |
| − | === '''''Notes: Because of the lack of the orthagonal symbol in the wikipedia formatting page, I will be type 'orth' in a superscript to symbolize that. ''''' === | + | === '''''Notes: Because of the lack of the orthagonal symbol in the wikipedia formatting page, I will be type 'orth' in a superscript to symbolize that. ''''' === |
| − | === '''<br>''' === | + | === '''<br>''' === |
| − | === | + | === '''''Analysis:''''' === |
| − | === '''First, let us say we have the following:''' === | + | === '''First, let us say we have the following:''' === |
| − | === '''V which is a subspace of R<sup>n</sup>, and {v<sub>1</sub>,v<sub>2</sub>,v<sub>3</sub>,..,v<sub>k</sub>} are a basis for V. (The entries in the braces are vectors)''' === | + | === '''V which is a subspace of R<sup>n</sup>, and {v<sub>1</sub>,v<sub>2</sub>,v<sub>3</sub>,..,v<sub>k</sub>} are a basis for V. (The entries in the braces are vectors)''' === |
| − | === '''To refresh, a basis means those entries span V, AND are also linearly independent. ''' === | + | === '''To refresh, a basis means those entries span V, AND are also linearly independent. ''' === |
| − | === '''<br>''' === | + | === '''<br>''' === |
| − | === '''So, therefore, then dim(V)=k (k is the number of vectors in our basis, which obviously is a non-finite amount, so I use k to denote that fact.)''' === | + | === '''So, therefore, then dim(V)=k (k is the number of vectors in our basis, which obviously is a non-finite amount, so I use k to denote that fact.)''' === |
| − | === '''<br>''' === | + | === '''<br>''' === |
| − | === '''Now that we have those assumptions and definitions out of the way, let me construct a matrix for you.''' === | + | === '''Now that we have those assumptions and definitions out of the way, let me construct a matrix for you.''' === |
| − | === '''<br>''' === | + | === '''<br>''' === |
=== '''We will call this matrix A (A seems to the most common letter in the linear algebra world...but i digress)''' === | === '''We will call this matrix A (A seems to the most common letter in the linear algebra world...but i digress)''' === | ||
| − | |||
| − | === '''Matrix A is a (n x k) matrix. (For reference, n is the rows, and k is the columns)''' === | + | === '''Matrix A is a (n x k) matrix. (For reference, n is the rows, and k is the columns)''' === |
=== '''<math>\begin{bmatrix} | === '''<math>\begin{bmatrix} | ||
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\end{bmatrix}</math>''' === | \end{bmatrix}</math>''' === | ||
| − | |||
| − | === '''Observe that the vectors are not just ordinary vectors, but rather, they are a very specific type called a COLUMN vector... === | + | === '''Observe that the vectors are not just ordinary vectors, but rather, they are a very specific type called a COLUMN vector... ''' === |
| − | === '''Pressing on, we can say the following:''' === | + | === '''Pressing on, we can say the following:''' === |
| − | === '''V= span(v<sub>1</sub>,v<sub>2</sub>,v<sub>3</sub>,..,v<sub>k</sub>)= c(A) (Again, the entries in the braces are vectors)''' === | + | === '''V= span(v<sub>1</sub>,v<sub>2</sub>,v<sub>3</sub>,..,v<sub>k</sub>)= c(A) (Again, the entries in the braces are vectors)''' === |
| − | === '''c(A) is a way to notate the column space of A. Additionally, look at the important relation that the span is equal to the column space of A! Keep this in mind, as it will be important for the rest of this. <br>''' === | + | === '''c(A) is a way to notate the column space of A. Additionally, look at the important relation that the span is equal to the column space of A! Keep this in mind, as it will be important for the rest of this. <br>''' === |
| − | === '''''The next statement is an axiom from the textbook, and should be implied (i.e. to proove the fundamentals of the statement is over my head).''''' === | + | === '''''The next statement is an axiom from the textbook, and should be implied (i.e. to proove the fundamentals of the statement is over my head).''''' === |
| − | === '''N(A<sup>T</sup>)= c(A<sup>orth</sup>)''' === | + | === '''N(A<sup>T</sup>)= c(A<sup>orth</sup>)''' === |
| − | === '''This states that the nullity of A transpose is equal to the column space of A's orthagonal compliment. ''' === | + | === '''This states that the nullity of A transpose is equal to the column space of A's orthagonal compliment. ''' === |
=== '''Now, that above relationship between nullity and the orthagonal compliment is straight from the textbook, but I will now relate that to the above work which has been done. ''' === | === '''Now, that above relationship between nullity and the orthagonal compliment is straight from the textbook, but I will now relate that to the above work which has been done. ''' === | ||
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| − | === ''' | + | === '''It is logical to say that ''' === |
| − | === ''' | + | === '''N(A<sup>T</sup>)= c(A<sup>orth</sup>) is ALSO equal to V<sup>orth</sup>. (i.e. if V=c(A), then it will apply if you '''orthagonalize'''both matrices as well).''' === |
| − | === ''' | + | === '''So, that leaves us with a nice triple equality, ''' === |
| + | === '''N(A<sup>T</sup>)= c(A<sup>orth</sup>)=V<sup>orth</sup>. Brilliant''' === | ||
| + | === '''<br>''' === | ||
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Revision as of 14:39, 8 December 2010
Contents
- 1
- 2 Statement: I am going to show that if V is a subspace of Rn. then dim(V)+dim(Vorth)=n
- 2.1 Notes: Because of the lack of the orthagonal symbol in the wikipedia formatting page, I will be type 'orth' in a superscript to symbolize that.
- 2.2
- 2.3 Analysis:
- 2.4 First, let us say we have the following:
- 2.5 V which is a subspace of Rn, and {v1,v2,v3,..,vk} are a basis for V. (The entries in the braces are vectors)
- 2.6 To refresh, a basis means those entries span V, AND are also linearly independent.
- 2.7
- 2.8 So, therefore, then dim(V)=k (k is the number of vectors in our basis, which obviously is a non-finite amount, so I use k to denote that fact.)
- 2.9
- 2.10 Now that we have those assumptions and definitions out of the way, let me construct a matrix for you.
- 2.11
- 2.12 We will call this matrix A (A seems to the most common letter in the linear algebra world...but i digress)
- 2.13 Matrix A is a (n x k) matrix. (For reference, n is the rows, and k is the columns)
- 2.14 $ \begin{bmatrix} v_{1} & v_{2} & v_{3} & v_{k} \\ v_{1} & v_{2} & v_{3} & v_{k} \\ v_{1} & v_{2} & v_{3} & v_{k} \\ .... & .... & .... & .... \\ .... & .... & .... & .... \\ .... & .... & .... & .... \\ v_{1} & v_{2} & v_{3} & v_{k} \\ \end{bmatrix} $
- 2.15 Observe that the vectors are not just ordinary vectors, but rather, they are a very specific type called a COLUMN vector...
- 2.16 Pressing on, we can say the following:
- 2.17 V= span(v1,v2,v3,..,vk)= c(A) (Again, the entries in the braces are vectors)
- 2.18 c(A) is a way to notate the column space of A. Additionally, look at the important relation that the span is equal to the column space of A! Keep this in mind, as it will be important for the rest of this.
- 2.19 The next statement is an axiom from the textbook, and should be implied (i.e. to proove the fundamentals of the statement is over my head).
- 2.20 N(AT)= c(Aorth)
- 2.21 This states that the nullity of A transpose is equal to the column space of A's orthagonal compliment.
- 2.22 Now, that above relationship between nullity and the orthagonal compliment is straight from the textbook, but I will now relate that to the above work which has been done.
- 2.23 It is logical to say that
- 2.24 N(AT)= c(Aorth) is ALSO equal to Vorth. (i.e. if V=c(A), then it will apply if you orthagonalizeboth matrices as well).
- 2.25 So, that leaves us with a nice triple equality,
- 2.26 N(AT)= c(Aorth)=Vorth. Brilliant
- 2.27
Statement: I am going to show that if V is a subspace of Rn. then dim(V)+dim(Vorth)=n
Notes: Because of the lack of the orthagonal symbol in the wikipedia formatting page, I will be type 'orth' in a superscript to symbolize that.
Analysis:
First, let us say we have the following:
V which is a subspace of Rn, and {v1,v2,v3,..,vk} are a basis for V. (The entries in the braces are vectors)
To refresh, a basis means those entries span V, AND are also linearly independent.
So, therefore, then dim(V)=k (k is the number of vectors in our basis, which obviously is a non-finite amount, so I use k to denote that fact.)
Now that we have those assumptions and definitions out of the way, let me construct a matrix for you.
We will call this matrix A (A seems to the most common letter in the linear algebra world...but i digress)
Matrix A is a (n x k) matrix. (For reference, n is the rows, and k is the columns)
$ \begin{bmatrix} v_{1} & v_{2} & v_{3} & v_{k} \\ v_{1} & v_{2} & v_{3} & v_{k} \\ v_{1} & v_{2} & v_{3} & v_{k} \\ .... & .... & .... & .... \\ .... & .... & .... & .... \\ .... & .... & .... & .... \\ v_{1} & v_{2} & v_{3} & v_{k} \\ \end{bmatrix} $
Observe that the vectors are not just ordinary vectors, but rather, they are a very specific type called a COLUMN vector...
Pressing on, we can say the following:
V= span(v1,v2,v3,..,vk)= c(A) (Again, the entries in the braces are vectors)
c(A) is a way to notate the column space of A. Additionally, look at the important relation that the span is equal to the column space of A! Keep this in mind, as it will be important for the rest of this.
The next statement is an axiom from the textbook, and should be implied (i.e. to proove the fundamentals of the statement is over my head).
N(AT)= c(Aorth)
This states that the nullity of A transpose is equal to the column space of A's orthagonal compliment.
Now, that above relationship between nullity and the orthagonal compliment is straight from the textbook, but I will now relate that to the above work which has been done.
It is logical to say that
N(AT)= c(Aorth) is ALSO equal to Vorth. (i.e. if V=c(A), then it will apply if you orthagonalizeboth matrices as well).
So, that leaves us with a nice triple equality,
N(AT)= c(Aorth)=Vorth. Brilliant
