Linear Algebra: quick little question about sums of subspaces

[tylerc1991]

Homework Statement

Learning about sums of subspaces and wanted to be sure that I am understanding this correctly. Say that I have two subspaces of R^2:

U = {(x,y) in R^2 : y + 2x = 0}
W = {(x,y) in R^2 : y - 3x = 0}

and I wanted to geometrically (and algebraically) represent their sum. What would it look like?

The Attempt at a Solution

Geometrically: since U is the line y = -2x and W is the line y = 3x, their sum, U + W, would be represented by both lines y = -2x and y = 3x appearing on one plane. But I am getting hung up on this new set, U + W, being closed even under addition. Ex: take the point (1,-2) from U and (1,3) from W, their sum is (2,1) which is not on either line y = -2x or y = 3x. What am I doing wrong here?

Algebraically: basing this off of the geometric interpretation, I would say that the sum of U + W would be represented by the set: {(x,y) in R^2 : y + 2x = 0 or y - 3x = 0}, but I am getting hung up on the same problems I was having with the geometric interpretation.

Thank you anyone for your help!

 

[micromass]

Hi tylerc1991!

What you seem to be doing is $U\cup W$, but like you notice, nobody says that this is closed uner addition, and it isn't.
What U+W actually is, is the space spanned by $U\cup W$. So you've correctly calculated that

$$U\cup W=\{(x,y)\in \mathbb{R}^2~\vert~y + 2x = 0~\text{or}~y - 3x = 0\}$$

but you still need to take the span of it.

To do this efficiently, take a basis of U, take a basis of W and take the space spanned by the union of these two bases.

Hope that helped!

 

[tylerc1991]

First, thank you for your help!

take a basis of U, take a basis of W and take the space spanned by the union of these two bases.

Would this not be the entire plane? My linear algebra is a little rusty , but I am thinking that the basis for U is some vector that points in the direction of the line y = -2x, and the basis for W is some other vector that points in the direction of the line y = 3x. The span of the union (addition) of these two vectors would be the whole plane.

 

[micromass]

Yep! We have

$$U+W=\mathbb{R}^2$$

Quite boring isn't it?

 

[tylerc1991]

Yep! We have

$$U+W=\mathbb{R}^2$$

Quite boring isn't it?

Haha, yes it isn't all that exciting. This was just an example I thought up to try to get a geometric meaning for the sum of subspaces. I was hoping that this wouldn't tangent into more questions (because of the title ), but it just might.

The definition of the sum of two subspaces given in the book I am using to brush up is this:
Given subspaces U_1, U_2, ..., U_m, the sum of these subspaces is denoted by U_1 + U_2 + ... + U_m and is defined as {u_1 + u_2 + ... + u_m : u_1 in U_1, u_2 in U_2, ..., u_m in U_m}. I hope this notation isn't confusing.

Using this definition, I am not seeing how we get R^2 in the example I am using.

 

[micromass]

Haha, yes it isn't all that exciting. This was just an example I thought up to try to get a geometric meaning for the sum of subspaces. I was hoping that this wouldn't tangent into more questions (because of the title ), but it just might.

The definition of the sum of two subspaces given in the book I am using to brush up is this:
Given subspaces U_1, U_2, ..., U_m, the sum of these subspaces is denoted by U_1 + U_2 + ... + U_m and is defined as {u_1 + u_2 + ... + u_m : u_1 in U_1, u_2 in U_2, ..., u_m in U_m}. I hope this notation isn't confusing.

Using this definition, I am not seeing how we get R^2 in the example I am using.

Excellent question. It is indeed true that U+W is defined as

$$U+W=\{u+w~\vert~u\in U, w\in W\}$$

However, the definition is not handy to work with. I prefer to use the equivalent

$$U+W=span(U\cup W)$$

because you can easily calculate this.

But let's get back to your question. Can we show, using the definition, that

$$U+W=\mathbb{R}^2$$

Yes! In fact, what we must do is for every $(x,y)\in \mathbb{R}^2$, we must find elements u and w of U and W such that u+w=(x,y).
Now, an arbitrary element of U has the form u=a(1,-2) for some a. Analogously, w is an element of W and thus has the form b(1,3).

So, we have to find a and b such that (x,y)=a(1,-2)+b(1,3). This yields a system of equations

$$\left\{\begin{array}{c} x=a+b\\ y=-2a+3b\\ \end{array}\right.$$

which you can easily solve.

For example, for (x,y)=(0,5), we have (0,5)=-(1,-2)+(1,3)=(-1,2)+(1,3). So we have expressed (0,5) as sum of vectors in U and W. We can do this with any vector by solving the above system.

 

[tylerc1991]

Ah, I see. Thank you micromass!