# Linear Algebra: Spanning sets and basis

Could you clarify what constitutes a spanning set and a basis? Also how does one test to see if a set of vectors is a spanning set and if it is a basis?

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#### Solution Preview

Let S be a set of vectors defined over certain field (say, the real numbers).

Now we can use these vectors to write a new one by linearly adding them together:

(1.1)

These new vectors form a vector space W

(1.2)

and the set S is called the "spanning set" of W.

For example, look at the set in :

(1.3)

Then all the vectors that can be created from these three vectors are of the form:

(1.4)

If we look at these vectors as Cartesian coordinates in real space, these vectors all lie in the xy plane.

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Not only that, any vector in the xy plane can be written as a linear combination of these vectors. Let be an arbitrary vector in the xy plane

(1.5)

So once we choose an arbitrary value for , equation (1.5) tells us what should be the values of the other two coefficients that will "manufacture" the vector w.

Hence the set is a spanning set of W.

Obviously, we can have many spanning sets for the same vector space

We could say that

(1.6)

Is also a spanning set of W (vectors in the xy plane)

(1.7)

again we see that we can write any vector in the xy plane as a linear combination of the vectors of equation (1.6) in this spanning set.

If we want to express the vector in terms of we get:

...

#### Solution Summary

The expert examines spanning sets and basis for linear algebra.