Confused with vector space concepts in general

Question

I'm having a really hard time grasping the concepts of the differences between some of the concepts having to do with vector spaces. Specifically what a spanning set is and what it means, as well as what a subspace is and how to find it. And if something does span something, what is the point of a linear combination? Does that have to do anything with linear independence?

Feeling a bit overwhelmed right now just not being able to grasp so many unfamiliar concepts. Thanks for any help you guys have.

Answer 1

The concept of spans and spanning sets is kind of a difficult concept for students to wrap their heads around when they first begin to study linear algebra (which is the study of vector spaces, NOT vectors).

There's some excellent videos on linear combinations and span available at Khan Academy. Follow this link.

https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/linear-combinations/v/linear-combinations-and-span

It's an excellent introductory video on linear combinations and spans. Sal Khan explains it in a very intuitive way. Check it out. I think you'll find it helpful.

Answer 2

Given a vectorspace $V$ over a field $K$, we know that $V$ is an abelian group. However, we also have scalar multiplication, meaning that if $v \in V$ and $\lambda \in K$ then $\lambda v$ is another element of $V$.

Since linear combinations of elements in $V$ are again in $V$, we can do the following: suppose we can describe $V$ by a subset $X$ of vectors. What does 'describe' mean in this context? It means that we can find any vector $v$ just by using the vectors in $X$, addition and scalar multiplication. This is exactly what a spanning set is: it is a set of vectors which completely describe a (sub)vectorspace. The description is found by addition and scalar multiplication of vectors in $X$ and this operation is called 'taking linear combinations of vectors in $X$'.

Note that it is always possible that a spanning set $X$ is 'too big' in the sense that we can describe exactly the same space by using less vectors. This means that the vectors in $X$ are linearly dependent. If we have a spanning set and this set is minimal (in the sense that removing one more vector no longer makes $X$ a spanning set), then $X$ is called a basis. One can prove that any two bases of the same vectorspace have the same cardinality.

Conclusion: Spanning sets gives us an easier way to represent vectorspaces. Each vector space $V$ has a spanning set, namely $V$ itself, but in most cases it is possible to pick a smaller set.

Hope this helps :)

Answer 3

A linear combination of two vectors $v_1$ and $v_2$ is simply adding or subtracting each of these vectors when multiplied by scalars.

For example,

If $v_1=\begin{pmatrix}6\\1\\-3\end{pmatrix},~ v_2=\begin{pmatrix}-3\\3\\2\end{pmatrix}$

Then $2v_1 + 3v_2$ = $2\begin{pmatrix}6\\1\\-3\end{pmatrix}+ 3\begin{pmatrix}-3\\3\\2\end{pmatrix}$ = $\begin{pmatrix}3\\11\\0\end{pmatrix}$

Here we are multiply $v_1$ by 2 and $v_2$ by 3 and simply add them together.

The vector $\begin{pmatrix}3\\11\\0\end{pmatrix}$ is a linear combination of the vectors $v_1,v_2$. But this is only one of the many linear combinations of the vectors $v_1,v_2$. There are many, many more.

For example, heres another linear combination of $v_1, v_2$

$v_1 - 2v_2$ = $\begin{pmatrix}6\\1\\-3\end{pmatrix}-2\begin{pmatrix}-3\\3\\2\end{pmatrix}$ = $\begin{pmatrix}12\\-5\\-7\end{pmatrix}$

The set of ALL such linear combination of $v_1$ and $v_2$ is called the span of $v_1$ and $v_2$ and is denoted $span(v_1,v_2)$

Hope this helps.

Answer 4

A "linear combination" of some vectors $v_1, \dots, v_n$ is just a fancy way of saying the following: it's the vector you get by multiplying the $v_1, \dots, v_n$ by some scalars and then adding them all together. So $2v_1 + v_2 - v_4 + \pi v_5$ is a linear combination of the vectors $v_1, \dots, v_8$.

Think about the following for a second. Points on a plane can be written by giving their coordinates, and you need two coordinates to specify a point: an x-coordinate and a y-coordinate. Thinking about this in a slightly different way, we can rephrase it in terms of vectors: given two preferred vectors in the x- and y-directions (one unit rightwards, and one unit upwards), the position vector of any point in the plane - say, $(3,5)$ - can be written as "3 units rightwards + 5 units upwards from the origin". This is a linear combination of those two vectors I mentioned!

Here's a natural question: given any vector space (maybe much bigger than a plane), can I somehow emulate the idea of "coordinates" on that vector space?

The following two concepts turn out to be really helpful:

• A set of vectors $v_1, \dots, v_n$ spans the vector space $V$ if every $x\in V$ can be written as a linear combination of them, say $x = a_1 v_1 + \dots + a_n v_n$, where $a_1, \dots, a_n$ are some scalars.
  • The point is: if $v_1, \dots, v_n$ is going to be a useful "coordinate system" for my vector space $V$, then I need to actually be able to get to every point in the space using them!
• A set of vectors $v_1, \dots, v_n$ is linearly independent if every vector that can be written as a linear combination of them can only be written as a linear combination in one unique way. That is, suppose we take two linear combinations: $a_1 v_1 + \dots + a_n v_n$ and $b_1 v_1 + \dots + b_n v_n$, and they happen to represent the same vector: $a_1 v_1 + \dots + a_n v_n = b_1 v_1 + \dots + b_n v_n$. If the set of vectors is linearly independent, these linear combinations must actually be the same one: that is, $a_1 = b_1, a_2 = b_2, \dots, a_n=b_n$.
  • The point is: if $v_1, \dots, v_n$ is going to be a useful "coordinate system" for my vector space $V$, I don't want one single point to have lots of different names.

That is:

• spanning sets are "big enough", in the sense that you can reach all vectors in the space using a spanning set.
• linearly independent sets are "small enough", in the sense that they contain no redundancy.

For example,

• the set of vectors $\left\{\begin{pmatrix} 1\\0\end{pmatrix}\right\}$ is not big enough to span the plane, but is linearly independent.
• the set of vectors $\left\{\begin{pmatrix} 1\\0\end{pmatrix}, \begin{pmatrix} 0\\1\end{pmatrix}\right\}$ is big enough to span the plane and is linearly independent.
• the set of vectors $\left\{\begin{pmatrix} 1\\0\end{pmatrix}, \begin{pmatrix} 0\\1\end{pmatrix}, \begin{pmatrix} 3\\5\end{pmatrix}, \begin{pmatrix} 10\\-1\end{pmatrix}\right\}$ is big enough to span the plane, but is not linearly independent.
• the set of vectors $\left\{\begin{pmatrix} 1\\0\end{pmatrix}, \begin{pmatrix} 2\\0\end{pmatrix}, \begin{pmatrix} 3\\0\end{pmatrix}, \begin{pmatrix} 576\\0\end{pmatrix}\right\}$ is not big enough to span the plane and is not linearly independent.

If a set is both "big enough" and "small enough" - i.e. it's "just the right size" - then we call it a basis. A basis does act exactly like a coordinate system on $V$, and bases have lots of cool properties. But if I write any more, I'll just be writing a textbook. :)