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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.

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Definition

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:

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Example 1

Under the definition given above, the *x-axis* is a subspace of **R**^{2}. 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.

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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 **R**^{3}.

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Spanning Sets and Subspaces

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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 **R**^{2}; 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 **R**^{2}? 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.