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Introduction To Subspaces Of Vector Spaces

Introduction To Subspaces Of Vector Spaces

Author: c o

To introduce the definition of a subspace
To define the concept of a span of vectors
To show that spans are subspaces

We take a look at subspaces and introduce the notion of the span of a set of vectors.

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The Definition Of A Subspace

Before starting this packet, you should have already understood the topics covered in this lesson introducing vector spaces.


Without being terribly exact, a subspace of a vector space is just what it sounds like - it is a "smaller" vector space inside of a "larger" vector space.  I use the words "smaller" and "larger" loosely because both can have (and in fact usually do have) infinitely many vectors within them.  

Ok, so imagine that we have one vector space that contains another - we don't necessarily have a subspace yet.  The additional requirement is that both spaces use the same definition of scalar multiplication and vector addition.  We put this additional requirement in because, as we shall see, there are times when the elements of one vector space are contained within another vector space, but the way addition or multiplication is defined differs.  

Enough chatter, we now define a subspace:

Example 1

Under the definition given above, the x-axis is a subspace of R2.  We see that adding any two "points" along the x-axis, which are just vectors that have a zero for the y-component, gives us another point on the x-axis.  Furthermore, we can multiply by constants, and the zero element is also present.



Example 2

Notice that above, the x-axis is really just all solutions to the homogeneous linear equation y=0.  That is, any point (x,0) will satisfy that equation.  In fact, the solutions to any homogeneous equation form a vector space.  Here we see a subspace of R3.

Spanning Sets and Subspaces

Definition of Span

For example, if S = {v} is just the set of a single vector, then [S] is just all multiples of that vector; i.e. [S] = {cv | c is in R}.  To be more explicit, the example of the x-axis.  The x-axis is just the span of the vector (1,0) in R2; it is everything that you can get by multiplying real numbers by that single vector (1,0).  But wait a minute! Didn't we see above that the x-axis is itself a subspace of R2?  As a matter of fact we did, this leads us to the reason we're interested in spans in the first place:

This means that we have a really easy way to find subspaces of vector spaces.  We just take a few vectors in the space, and take linear combinations of all of them, and wham-o, we have a subspace. 

Up next we will see a few examples of spans.

Examples of Spanning Sets

In this video we see a few examples of spanning sets, and we do some of the work required to show that they are vector spaces.

ERRATA: I kept using the word "constant" when I meant to say "scalar".