Question on subspaces and spans of vector spaces

[juanma101285]

Hi, I have read my notes and understand the theory, but I am having trouble understanding the following questions which are already solved (I am giving the answers as well).

The first question says:

Let $U_{1}$ and $U_{2}$ be subspaces of a vector space V. Give an example (say in $V=\Re^{2}$) to show that the union $U_{1}\bigcup U_{2}$ need not be a subspace of V.

And the answer is:

Take $U_{1}=\{(x_{1}, x_{2})\in\Re^{2}:x_{2}=0\}$ and $U_{2}=\{(x_{1}, x_{2})\in\Re^{2}:x_{1}=0\}$

So I really don't understand this answer... I would really appreciate it if someone could explain it to me.


And the second question says:

Let S be the set of all vectors $(x_{1}, x_{2})$ in $\Re^{2}$ such that $x_{1}=1$. What is the span of S?

And the answer is:

span S = $\Re^{2}$ because $(x_{1}, x_{2})=x_{1}(1, x^{-1}_{1}x_{2})$ when $x_{1}\neq 0$ and $(x_{1}, x_{2})=(1, 0)-(1, -x_{2})$ when $x_{1}=0$.

So I don't understand this explanation, mainly because I thought that $x_{1}$ is supposed to be 1... :/

Sorry if the questions sound silly, and thanks for any help you can give me! :)

 

[micromass]

For the first one: what don't you understand?? The solution claims three things:

1) U₁ is a subspace of $\mathbb{R}^2$
2) U₂ is a subspace of $\mathbb{R}^2$
3) $U_1\cup U_2$ is not a subspace of $\mathbb{R}^2$

Which of these three points is bothering you??

For the second one: A typical vector in S is (1,a) for a arbitrary. We are interesting in whether the span of S is $\mathbb{R}^2$.
The span is all linear combinations of elements of S. Thus if we claim that the span of S is $\mathbb{R}^2$, then we actually claim that each vector in $\mathbb{R}^2$ is a linear combination of elements in S. Can you find such a linear combination?

 

[Fredrik]

It looks like you haven't really studied the definitions of the terms you're using, in particular "span". The span of S is the set of all linear combinations of members of S. So it's almost obvious that span S=ℝ². To prove it, it's sufficient to show that an arbitrary member of ℝ² can be written as ax+by, where a and b are real numbers, and x and y are members of S. (Actually, this proves that ℝ² is a subset of span S, but since it's obvious that span S is a subset of ℝ², this is what we need to conclude that span S=ℝ²).

The first question makes me wonder if you understand the terms "union" and "subspace". It should be pretty obvious that the union isn't closed under the vector space operations (addition and scalar multiplication).

(I actually wrote this before micromass wrote his reply above, but my brother who is visiting me this week pulled me away from the computer for a while).

 

[juanma101285]

Thanks a lot, I feel like a retard now... I got very confused with the concept of subspace. I do understand it now though and it seems easy, so all the other problems make sense :)

 

[blue_raver22]

I understand that it can be difficult to grasp certain concepts, even after reading notes and explanations. I will do my best to explain the answers to these questions in a clear and understandable way.

For the first question, we are given two subspaces, U_{1} and U_{2}, of a vector space V. The question asks us to give an example where the union of these subspaces, U_{1}\bigcup U_{2}, is not a subspace of V. In order to understand this, we need to remember the definition of a subspace. A subspace is a subset of a vector space that is closed under addition and scalar multiplication. This means that if we take any two vectors from the subspace and add them together, the resulting vector must also be in the subspace. Similarly, if we take any vector from the subspace and multiply it by a scalar, the resulting vector must also be in the subspace.

Now, let's look at the example given in the answer. U_{1} is the set of all vectors in \Re^{2} where the second component, x_{2}, is equal to 0. Similarly, U_{2} is the set of all vectors in \Re^{2} where the first component, x_{1}, is equal to 0. In other words, U_{1} is the x-axis and U_{2} is the y-axis in the Cartesian plane. It is easy to see that both U_{1} and U_{2} are subspaces of \Re^{2}. However, when we take the union of these two subspaces, we get all the points on the x-axis and all the points on the y-axis. This means that the union, U_{1}\bigcup U_{2}, is the entire plane \Re^{2}. And since \Re^{2} is not closed under addition (think about what happens when we add two points on the x-axis or two points on the y-axis), the union is not a subspace of \Re^{2}. This example shows that the union of two subspaces may not always be a subspace.

Moving on to the second question, we are given a set S which consists of all vectors (x_{1}, x_{2}) in \Re^{2} such that x_{1}=1. The question asks us to find the span of this set,