Is A Closed Under Scalar Multiplication in W?

[bugatti79]

Is the subset A a subspace of W

W=\left \{ \begin{bmatrix}<br /> 1 &amp;1 \\ <br /> a_{21}&amp; a_{22} \\ <br /> a_{31}&amp; a_{32} <br /> \end{bmatrix} :a_{ij} \in \mathbb{C}\right \}

Let A=\begin{bmatrix}<br /> 1 &amp;1 \\ <br /> a_{21}&amp; a_{22}\\ <br /> a_{31}&amp; a_{32} <br /> \end{bmatrix}

A \in W

Then 2A \in W since

2A=2\begin{bmatrix}<br /> 1 &amp;1 \\ <br /> a_{21}&amp; a_{22}\\ <br /> a_{31}&amp; a_{32} <br /> \end{bmatrix}

Is this correct? My notes is telling its NOT closed under scalar multiplication which I don't think is correct.

 

[Ryker]

What exactly is A, a matrix? Or what do the two lines around the components mean? In any case, multiply 2A and write it out, so that you have it in the form with the two lines around it. You should then be able to see why this isn't a subspace.

(Unless those two lines stand for something else than what I had in mind.)

 

[Mark44]

2A is NOT in W since the entries in the top row aren't both 1.

Ryker, A is a matrix. I changed the OP's LaTeX so that it didn't look like a determinant.

 

[bugatti79]

2A is NOT in W since the entries in the top row aren't both 1.

Ryker, A is a matrix. I changed the OP's LaTeX so that it didn't look like a determinant.

Thanks for the clarification on the lines.

Ok, so this is true for any scalar \alpha \in \mathbb{R}?

 

[Ryker]

No, it's not true for any scalar (it is true for all but one - can you tell, which one?), but to show this subset is not a subspace, you only need to find one scalar where this fails.

 

[bugatti79]

Now I am confused...
Based on post #3, it is NOT a subspace because the 1's are not present in the matrix due to multiplication of real scalar '2'. I thought this WOULD be true for \alpha \in (0,\infty) where the round brackets mean excluding 0 and infinity.

But you are saying its not true for all real scalars...?!

 

[Mark44]

No, it's not true for any scalar (it is true for all but one - can you tell, which one?),

I think you mean that it's true for only one scalar.

but to show this subset is not a subspace, you only need to find one scalar where this fails.

 

[Mark44]

Now I am confused...
Based on post #3, it is NOT a subspace because the 1's are not present in the matrix due to multiplication of real scalar '2'. I thought this WOULD be true for \alpha \in (0,\infty) where the round brackets mean excluding 0 and infinity.

But you are saying its not true for all real scalars...?!

What post #3 is saying is that, in general, if A \in W, then cA \notin W, so scalar multiplication is not closed in W.

BTW, why are you limiting you scalars to positive real numbers? Set W has matrices with entries from C, the field of complex numbers. Any scalars would also come from this field.

 

[bugatti79]

What post #3 is saying is that, in general, if A \in W, then cA \notin W, so scalar multiplication is not closed in W.

BTW, why are you limiting you scalars to positive real numbers? Set W has matrices with entries from C, the field of complex numbers. Any scalars would also come from this field.

Ok,

TO clarify...the subset is not a subspace for all scalars \in \mathbb{C} except the scalar '1'...that it?

 

[Ryker]

I think you mean that it's true for only one scalar.

I thought by "is this true for all scalars" he meant "is it true for all scalars a, that aA is not an element of W" So this would be true for all scalars, except for one, for which it would hold that aA is in W. So much confusion

 

[Ryker]

Ok,

TO clarify...the subset is not a subspace for all scalars \in \mathbb{C} except the scalar '1'...that it?

No, you can't say the subset is a subspace for all scalars except for one. Either it is a subspace or it isn't.

 

[bugatti79]

No, it's not true for any scalar (it is true for all but one - can you tell, which one?), but to show this subset is not a subspace, you only need to find one scalar where this fails.

ok, which scalar makes A a subspace...? It can't be 0, i, infinity...I haven't a clue...?

 

[Mark44]

None, and that's the point of this exercise. A is NOT a subspace of the space of 3 X 2 matrices with entries from C.

Set A is not closed under scalar multiplication. This set is also not closed under addition.

 

[bugatti79]

ok, cheers!

 

[Ryker]

As mentioned above, by "it is true for all but one", I meant that multiplying A by a particular scalar (call it "a"), will result in an element of W, i.e. aA is in W, but if you multiply A by any other scalar, say, "b", bA is not in W.

Other than that, perhaps revisit the definition of a subspace. No scalar can make a subspace, since being a subspace is a property of the set itself. So if scalar multiplication with an element of your set results in another element of your set, then the set is closed under scalar multiplication. But if you can find one (or two or a thousand or infinitely many) scalar for which this statement is not true, then your set is not closed under scalar multiplication.

 

[bugatti79]

As mentioned above, by "it is true for all but one", I meant that multiplying A by a particular scalar (call it "a"), will result in an element of W, i.e. aA is in W, but if you multiply A by any other scalar, say, "b", bA is not in W.

but what is so special about 'a' that makes it a subspace when 'alpha' a scalar entry from C doesnt?

 

[Ryker]

Just to clear up confusion, I used "a" instead of your "alpha", because I didn't want to go into tex mode. But a = alpha, I just used a different variable name. So in my replies, a is also an element of C, i.e. a is a complex number, a scalar. And the whole point of my replies was that that special "a" or "alpha" does NOT make your set a subspace. That is, if your set is closed under multiplication by a particular scalar, then this is needed, but not sufficient for a set to be a subspace, because it needs to be closed under multiplication by all scalars. But if you do find one scalar, under multiplication by which your set is not closed, then that is sufficient in showing your set is not a subspace.

 

[Mark44]

Nothing, and the name of the scalar has no bearing on things. For scalar multiplication to be closed, aA has to be in the set for every scalar in whatever field is relevant to the problem.

 

[bugatti79]

I am looking at my notes and found a counter example...!

In R^3 where the vector x = (1,2,-4)

apparently its closed under addition if we add it to another vector y etc...but 2x=(2,4,-8)...according to this post...this would imply x is NOT a subspace (because 1,2 and -4 is not in the new set)...but notes say it is a real vector space...!?

 

[Ryker]

What do you mean? 2x is still in R^3, so everything is as it should be.

 

[bugatti79]

What do you mean? 2x is still in R^3, so everything is as it should be.

according to post #3, 2A is NOT a subspace...im just following what I interpret from this thread...because the entries are not 1,2,-4 anymore...

 

[Ryker]

Ugh, I'm sorry to say this, but you really do need to go back to your notes, read the definitions and let them sink in. We were talking about [STRIKE]W[/STRIKE]A being a subspace, not 2A. A is just an element in W (and A), whereas as we have shown, 2A is not.

edit: Correction, we were talking about A being a subspace, and we have clumsily denoted entries in A as A's, as well. And then we have shown that 2A, where A is an element of A (ugh), is not in A (where as it the set A, not the element A).

 

[bugatti79]

aha...I believe I understand you now. The only way for 2A to be an element of W is if we actually redefine W such that (purely for demonstration purposes)

W=\left \{ \begin{bmatrix}<br /> b_{11} &amp;b_{12} \\ <br /> 0&amp; 0 \\ <br /> 0&amp; 0 <br /> \end{bmatrix} :b_{ij} \in{R}\right \}...

 

[Ryker]

What would A now be?

 

[bugatti79]

What would A now be?

and A would be A=\begin{bmatrix}<br /> 1 &amp;1 \\ <br /> 0&amp;0\\ <br /> 0&amp;0 <br /> \end{bmatrix}

 

[Ryker]

Yes, then if A as the set (and not the element matrix) is the set of all A's (matrices this time), and W is redefined as you have suggested, 2A would be in W.

 

[Deveno]

one of the problems with this problem, is a failure to clearly define what the vector space W and the subset A ARE.

this is what i think is intended:

W is the space of all 3x2 matrices with complex entries (over the complex field).

A is the subset of all 3x2 matrices with all 1's in the first row.

is A a subspace of W?

THAT, is a well-defined question.

for A to be a subspace of W, it has to be a vector space in its own right (using the "inherited" vector addition and scalar multiplication of W). this means, in particular 3 things:

1). A cannot be empty (the empty set is NOT a vector space).
2) the vector sum on W must yield a well-defined binary operation on A:

+:AxA→A

that is, if u,v are in A, u+v must be as well (this is called CLOSURE of vector addition, and must be true of any vector space).

3)scalar multplication must yield a well defined operation:

* :FxA→A (typically one writes cu instead of the cumbersome *(c,u)).

that is, if c is any scalar in the underlying field F (F is the complex numbers, in this problem), then for any element u in A, cu must be in A, as well.

(this is called closure of scalar multiplication, and again, ANY vector space, not just subspaces, must satisfy this rule).

since the other axioms for a vector space only depend on the behavior of the operations, if the parent space already satisfies them, we don't have to check the subspace (any element of A is automatically an element of W) for the commutativity of addition, the distributive laws, etc. but these 3 rules are ESSENTIAL.

in the problem at hand, A is not a subspace. we're ok on rule (1), A is obviously not empty. but we run into trouble with both (2) AND (3). for example:

P = \begin{bmatrix}1&amp;1\\0&amp;0\\0&amp;0\end{bmatrix} \in A,\ Q = \begin{bmatrix}1&amp;1\\1&amp;1\\0&amp;0\end{bmatrix} \in A

however, the sum:

P+Q = \begin{bmatrix}2&amp;2\\1&amp;1\\0&amp;0\end{bmatrix} \not \in A

since the first row is not all 1's. it's also clear that cP is not in A, unless c = 1.

 

[bugatti79]

Thanks Deveno. Your notes are valuable and is a supplement to my notes. Do you know what has happened to Mathhelpforum?