Linear Algebra Subspace

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KindofSlow

Junior Member
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Mar 5, 2010
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Hello All,
Book (Linear Algebra by Jim Hefferon) says this set is not asubspace of R2.



X
0



1
0


{( ) + ( ) │x ϵ R}

Seems to me it contains zerovector and is closed with respect to both addition and scalar multiplication,making it a subspace. Also seems like this set is the x axis which would also makeit a subspace.

As always, any insightsteering me away from misunderstandings is greatly appreciated.

Thank you

 
I'm not clear what you are asking. Are your vectors of the form \(\displaystyle \begin{pmatrix}x \\ 0 \end{pmatrix}\)? if so, what is the \(\displaystyle \begin{pmatrix}1 \\ 0 \end{pmatrix}\) shown for and what does "\(\displaystyle \{( )+ ()| x\in R\}\)" mean?

Are the operations "component wise" addition and scalar multiplication?
 
My apologies for the formatting, it looked good until I hit "Submit..." and then it got messed up.
The (1,0) and (x,0) are supposed to be two little column vectors in between the two () () below, so yes what you put are the two vectors.
Make sense?
Thank you
 
HallsofIvy said:
I'm not clear what you are asking. Are your vectors of the form \(\displaystyle \begin{pmatrix}x \\ 0 \end{pmatrix}\)? if so, what is the \(\displaystyle \begin{pmatrix}1 \\ 0 \end{pmatrix}\) shown for and what does "\(\displaystyle \{( )+ ()| x\in R\}\)" mean?

Are the operations "component wise" addition and scalar multiplication?
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Professor, unless otherwise stated are we to assume that the scalars are in R?
 
Yes, scalars are in R.
Also, I am very aware that I don't know what the heck I'm doing so if my question is too incoherent to answer, just let me know and I'll carry on until I am knowledgeable enough to ask a lucid question.
Thank you
 
KindofSlow said:
Yes, scalars are in R.
Also, I am very aware that I don't know what the heck I'm doing so if my question is too incoherent to answer, just let me know and I'll carry on until I am knowledgeable enough to ask a lucid question.
Thank you
Click to expand...
It seems to be a subspace of R2 as you mentioned (you called it the x-axis- actually it is the projection of R2 onto the x-axis). I want you to use the definition of subspace and write up the proof to show that it is a subspace. See if you get stuck anywhere showing that it is not in fact a subspace.
 
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KindofSlow said:
Book (Linear Algebra by Jim Hefferon) says this set is not a subspace of R2.

. . . . .\(\displaystyle \left\{\left(\begin{array}{c}x\\0\end{array}\right)\, +\, \left(\begin{array}{c}1\\0\end{array}\right)\, \bigg|\, x\, \in\, \mathbb{R}\right\}\)

Seems to me it contains zero vector and is closed with respect to both addition and scalar multiplication, making it a subspace. Also seems like this set is the x axis which would also make it a subspace.
Click to expand...
Does the book give any other information about this set? Because, as currently posted, I'm not seeing how this isn't a subspace. I mean, it's clearly not the empty set, and:

. . . . .\(\displaystyle \left(\begin{array}{c}-1\\0\end{array}\right)\, +\, \left(\begin{array}{c}1\\0\end{array}\right)\, =\, \left(\begin{array}{c}0\\0\end{array}\right)\,\)

. . . . .\(\displaystyle \left(\begin{array}{c}x\\0\end{array}\right)\, +\, \left(\begin{array}{c}y\\0\end{array}\right)\, =\, \left(\begin{array}{c}x\, +\, y\, -\, 1\\0\end{array}\right)\, +\, \left(\begin{array}{c}1\\0\end{array}\right)\,\)

. . . . .\(\displaystyle a\, \cdot\, \left(\begin{array}{c}x\\0\end{array}\right)\, =\, \left(\begin{array}{c}ax\, -\, 1\\0\end{array}\right)\, +\, \left(\begin{array}{c}1\\0\end{array}\right)\,\)

What are we missing? :oops:
 
stapel said:
Does the book give any other information about this set? Because, as currently posted, I'm not seeing how this isn't a subspace. I mean, it's clearly not the empty set, and:

. . . . .\(\displaystyle \left(\begin{array}{c}-1\\0\end{array}\right)\, +\, \left(\begin{array}{c}1\\0\end{array}\right)\, =\, \left(\begin{array}{c}0\\0\end{array}\right)\,\)

. . . . .\(\displaystyle \left(\begin{array}{c}x\\0\end{array}\right)\, +\, \left(\begin{array}{c}y\\0\end{array}\right)\, =\, \left(\begin{array}{c}x\, +\, y\, -\, 1\\0\end{array}\right)\, +\, \left(\begin{array}{c}1\\0\end{array}\right)\,\)

. . . . .\(\displaystyle a\, \cdot\, \left(\begin{array}{c}x\\0\end{array}\right)\, =\, \left(\begin{array}{c}ax\, -\, 1\\0\end{array}\right)\, +\, \left(\begin{array}{c}1\\0\end{array}\right)\,\)

What are we missing? :oops:
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I don't think anything is missing. Your proof above is all that is needed. To the op, can you please upload the page in your textbook that says this is not a subspace?
 
Hello Everyone,
Sorry for the long delay but I wanted to get this figured out before replying.
I have confirmed definitely this is a subspace and the book is wrong.
Thank you all very much for all your assistance.
 
KindofSlow said:
Hello Everyone,
Sorry for the long delay but I wanted to get this figured out before replying.
I have confirmed definitely this is a subspace and the book is wrong.
Thank you all very much for all your assistance.
Click to expand...
Hi, I know that there are errors in textbook but this one troubles me. If you have the time can you please post the page(s) from your book regarding this problem, Thanks!
 
Jomo said:
Hi, I know that there are errors in textbook but this one troubles me. If you have the time can you please post the page(s) from your book regarding this problem, Thanks!
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He has posted the textbook and page number in this thread. I found it but the page 90 is blacked-out. Frankly, I have never seen the book even though it was published before I actually left the department for good. I suspect that notation means something else other that span as we have been reading it. Anyone writing a LA textbook should know that those two vectors are dependent and spam the same space. So I think there is surely more to it than we see. I cannot do this, but I wish someone would email that author to make him aware of this thread and ask him to respond.

PS Book (Linear Algebra by Jim Hefferon) says this set is not a subspace of R2.

James Hefferon.
 
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pka said:
He has posted the textbook and page number in this thread. I found it but the page 90 is blacked-out. Frankly, I have never seen the book even though it was published before I actually left the department for good. I suspect that notation means something else other that span as we have been reading it. Anyone writing a LA textbook should know that those two vectors are dependent and spam the same space. So I think there is surely more to it than we see. I cannot do this, but I wish someone would email that author to make him aware of this thread and ask him to respond.

PS Book (Linear Algebra by Jim Hefferon) says this set is not a subspace of R2.

James Hefferon.
Click to expand...
Contact has been made.
 
Hefferon got back to me via an email. He said that he does not see the example in his book which was originally posted. The book is available online and when I looked at it, it does not have the problem. The book can be viewed at http://joshua.smcvt.edu/linearalgebra/book.pdf
It would be most helpful if the OP would supply us with a picture of the page. I plan on sending the OP a message requesting it.
 
Hello Everyone,
Wow, I apologize as I feel like I created a firestorm here and that was not my intent.
I'll try to clarify everything here.
My question came from Exercise 4.20 (e) on page 135 of the book PDF (Chapter 2 (Vector Spaces), Section III.4 (Basis and Dimension . Combining Subspaces).
The answer came from the separate "Answers" PDF that goes with the textbook - Two.III.4.20 on page 68.
Full question is "Decide if R2 is the direct sum of each W1 and W2. (e) W1 is {(1,0)+(x,0)} and W2 is {(-1,0)+(0,y)} (both x and y real numbers).
The answer in the Answers PDF is "(e) No. These are not subspaces."
I interpreted this to mean "Neither of these is a subspace", which led me to post my original question.
I have since been informed that another interpretation of the answer could be " "Both of these are subspaces" is a false statement." Which, of course, would be a correct answer.

You are all really great and very helpful and I hope you'll let me keep posting questions and I'll try to not be such a troublemaker.

Thank you very much for all your assistance.
 
KindofSlow said:
Hello Everyone,
Wow, I apologize as I feel like I created a firestorm here and that was not my intent.
I'll try to clarify everything here.
My question came from Exercise 4.20 (e) on page 135 of the book PDF (Chapter 2 (Vector Spaces), Section III.4 (Basis and Dimension . Combining Subspaces).
The answer came from the separate "Answers" PDF that goes with the textbook - Two.III.4.20 on page 68.
Full question is "Decide if R2 is the direct sum of each W1 and W2. (e) W1 is {(1,0)+(x,0)} and W2 is {(-1,0)+(0,y)} (both x and y real numbers).
The answer in the Answers PDF is "(e) No. These are not subspaces."
I interpreted this to mean "Neither of these is a subspace", which led me to post my original question.
I have since been informed that another interpretation of the answer could be " "Both of these are subspaces" is a false statement." Which, of course, would be a correct answer.

You are all really great and very helpful and I hope you'll let me keep posting questions and I'll try to not be such a troublemaker.

Thank you very much for all your assistance.
Click to expand...
By any chance is your name Denis? (no need to reply. It's just an inside job about one of the volunteers)
 
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