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| − | + | <span class="texhtml">''I''''n''''s''''e''''r''''t''''f''''o''''r''''m''''u''''l''''a''''h''''e''''r''''e''</span> | |
| − | ==Homework 1 collaboration area== | + | == Homework 1 collaboration area == |
| − | Feel free to toss around ideas here. | + | Feel free to toss around ideas here. Feel free to form teams to toss around ideas. Feel free to create your own workspace for your own team. --[[User:Bell|Steve Bell]] 12:11, 20 August 2010 (UTC) |
| − | Here is my favorite formula: | + | Here is my favorite formula: |
| − | <math>f(a)=\frac{1}{2\pi i}\int_\gamma \frac{f(z)}{z-a}\ dz.</math> | + | <math>f(a)=\frac{1}{2\pi i}\int_\gamma \frac{f(z)}{z-a}\ dz.</math> |
| − | Question from a student: | + | Question from a student: |
| − | I have a question about P.301 #33. | + | I have a question about P.301 #33. I see in the back of the book that it is not a vector space. I don't understand why though. In the simplest form I would think an identity matrix would satisfy the requirements mentioned in the book for #33. Isn't an identity matrix a vector space? |
| − | not a vector space. | + | |
| − | would think an identity matrix would satisfy the requirements mentioned in | + | |
| − | the book for #33. | + | |
| − | Answer from Bell: | + | Answer from Bell: |
| − | One of the key elements of being a vector space is | + | One of the key elements of being a vector space is that the thing must be closed under addition. The set consisting of just the identity matrix is not a vector space because if I add the identity matrix to itself, I get a matrix with twos down the diagonal, and that isn't in the set. So it isn't closed under addition. (The set consisting of the ZERO matrix is a vector space, though.) |
| − | that the thing must be closed under addition. | + | |
| − | set consisting of just the identity matrix is not | + | |
| − | a vector space because if I add the identity matrix | + | |
| − | to itself, I get a matrix with twos down the diagonal, | + | |
| − | and that isn't in the set. | + | |
| − | addition. | + | |
| − | is a vector space, though.) | + | |
| − | Another question from the same student: | + | Another question from the same student: |
| − | Does a set of vectors have to contain the 0 vector to be a vector space? | + | Does a set of vectors have to contain the 0 vector to be a vector space? If that is true, then any set of vectors that are linearly independent would not be a vector space? |
| − | Answer from Bell: | + | Answer from Bell: |
| − | Yes, a vector space must contain the zero vector | + | Yes, a vector space must contain the zero vector (because a constant times any vector has to be in the space if the vector is, even if the constant is zero). |
| − | (because a constant times any vector has to be in | + | |
| − | the space if the vector is, even if the constant | + | |
| − | is zero). | + | |
| − | The set of all linear combinations of a bunch of | + | The set of all linear combinations of a bunch of vectors is a vector space, and the zero vector is in there because one of the combinations involves taking all the constants to be zero. |
| − | vectors is a vector space, and the zero vector is | + | |
| − | in there because one of the combinations involves | + | |
| − | taking all the constants to be zero. | + | |
| − | Question from a student: | + | Question from a student: |
| − | How do I start problem 6 (reducing the matrix) on page 301? I understand that the row and column space will be the same, but I'm not sure how to get it to reduce to row-echelon form. | + | How do I start problem 6 (reducing the matrix) on page 301? I understand that the row and column space will be the same, but I'm not sure how to get it to reduce to row-echelon form. |
| − | Answer from Bell: | + | Answer from Bell: |
| − | You'll need to do row reduction. | + | You'll need to do row reduction. At some point you'll get |
| − | At some point you'll get | + | <pre> 1 1 a |
| − | + | ||
| − | < | + | |
| − | + | ||
0 a-1 1-a | 0 a-1 1-a | ||
0 1-a 1-a^2 | 0 1-a 1-a^2 | ||
| − | </ | + | </pre> |
| + | At this point, you'll need to consider the case a=1. If a=1, the matrix is all 1's and the row space is just the linear span of | ||
| − | + | [ 1 1 1 ]. | |
| − | + | ||
| − | + | ||
| − | + | If a is not 1, then you can divide row two by a-1 and continue doing row reduction. I think you'll have one more case to consider when you try to deal with the third row after that. | |
| − | + | [Q: Only 1 more case? Don't we need to consider the case in which a is not equal to 1 and a is not equal to the value found from the third row after we continue row reduction? ] | |
| − | + | ||
| − | + | ||
| − | to consider | + | |
| − | after | + | |
| − | Question from student: | + | Question from student: |
| − | Also, what are you wanting for an answer to questions 22-25 (full proof, explanation, example, etc.)? | + | Also, what are you wanting for an answer to questions 22-25 (full proof, explanation, example, etc.)? |
| − | Anwer from Bell: | + | Anwer from Bell: |
| − | For problems 22-25, I'd want you to explain the answer | + | For problems 22-25, I'd want you to explain the answer so that someone who has just read 7.4 would understand you and believe what you say. For example, the answer to #25 might start: |
| − | so that someone who has just read 7.4 would understand | + | |
| − | you and believe what you say. | + | |
| − | to #25 might start: | + | |
| − | If the row vectors of a square matrix are linearly | + | If the row vectors of a square matrix are linearly independent, then Rank(A) is equal to the number of rows. But Rank(A)=Rank(A^T). Etc. |
| − | independent, then Rank(A) is equal to the number | + | |
| − | of rows. | + | [[Category:MA5272010Bell]] |
Revision as of 19:48, 30 August 2010
I'n's'e'r't'f'o'r'm'u'l'a'h'e'r'e
Homework 1 collaboration area
Feel free to toss around ideas here. Feel free to form teams to toss around ideas. Feel free to create your own workspace for your own team. --Steve Bell 12:11, 20 August 2010 (UTC)
Here is my favorite formula:
$ f(a)=\frac{1}{2\pi i}\int_\gamma \frac{f(z)}{z-a}\ dz. $
Question from a student:
I have a question about P.301 #33. I see in the back of the book that it is not a vector space. I don't understand why though. In the simplest form I would think an identity matrix would satisfy the requirements mentioned in the book for #33. Isn't an identity matrix a vector space?
Answer from Bell:
One of the key elements of being a vector space is that the thing must be closed under addition. The set consisting of just the identity matrix is not a vector space because if I add the identity matrix to itself, I get a matrix with twos down the diagonal, and that isn't in the set. So it isn't closed under addition. (The set consisting of the ZERO matrix is a vector space, though.)
Another question from the same student:
Does a set of vectors have to contain the 0 vector to be a vector space? If that is true, then any set of vectors that are linearly independent would not be a vector space?
Answer from Bell:
Yes, a vector space must contain the zero vector (because a constant times any vector has to be in the space if the vector is, even if the constant is zero).
The set of all linear combinations of a bunch of vectors is a vector space, and the zero vector is in there because one of the combinations involves taking all the constants to be zero.
Question from a student:
How do I start problem 6 (reducing the matrix) on page 301? I understand that the row and column space will be the same, but I'm not sure how to get it to reduce to row-echelon form.
Answer from Bell:
You'll need to do row reduction. At some point you'll get
1 1 a 0 a-1 1-a 0 1-a 1-a^2
At this point, you'll need to consider the case a=1. If a=1, the matrix is all 1's and the row space is just the linear span of
[ 1 1 1 ].
If a is not 1, then you can divide row two by a-1 and continue doing row reduction. I think you'll have one more case to consider when you try to deal with the third row after that.
[Q: Only 1 more case? Don't we need to consider the case in which a is not equal to 1 and a is not equal to the value found from the third row after we continue row reduction? ]
Question from student:
Also, what are you wanting for an answer to questions 22-25 (full proof, explanation, example, etc.)?
Anwer from Bell:
For problems 22-25, I'd want you to explain the answer so that someone who has just read 7.4 would understand you and believe what you say. For example, the answer to #25 might start:
If the row vectors of a square matrix are linearly independent, then Rank(A) is equal to the number of rows. But Rank(A)=Rank(A^T). Etc.
