Let **u** and **v** be (fixed) vectors in the vector space *V*. Show that the set *W* of all linear combinations *a***b**+*b***v** of **u** and **v** is a subspace of *V*.

Show that generic elements of the set fulfill the definition of a vector space. (There should be a list of properties for vector spaces. You need to show that these linear combinations obey these rules.)

Prove: If the (finite) set *S* of vectors contains the zero vector, then *S* is linearly dependent.

What is the definition of a linearly independent set? If you add the zero vector to such a set, what property no longer holds?

Determine whether or not the given vectors in **R**^{n} form a basis for **R**^{n}.

**v**_{1}=(3,-7,5,2), **v**_{2}=(1,-1,3,4), **v**_{3}=(7,11,3,13)

Since the vectors are in R

^{4} and you have only three vectors (so clearly they cannot form a basis), try to find a vector in R

^{4} which is

*not* in the span of the set you've been given.

Let {**v**_{1},**v**_{2},...,**v**_{k}} be the basis for the proper subspace *W* of the vector space *V*, and suppose that the vector **v** of *V* is not in *W*. Show that the vectors **v**_{1},**v**_{2},...,**v**_{k},**v** are linearly independent.

What is the definition of a basis? So what can you say about the vectors in the basis?

If the vector

**v** is not in W, can you form

**v** by a linear combination of the basis vectors? So what can you say about the set with this new vector thrown in?