We are doing vector spaces in my math class right now and I'm TOTALLY lost. Here are a few of the problems I don't get:
Let u and v be (fixed) vectors in the vector space V. Show that the set W of all linear combinations ab+bv of u and v is a subspace of V.
Prove: If the (finite) set S of vectors contains the zero vector, then S is linearly dependent.
Determine whether or not the given vectors in Rn form a basis for Rn.
v1=(3,-7,5,2), v2=(1,-1,3,4), v3=(7,11,3,13)
Let {v1,v2,...,vk} 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 v1,v2,...,vk,v are linearly independent.
Please help! Thanks!
I really need help on this stuff
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isuckatmath
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stapel_eliz
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isuckatmath wrote:Let u and v be (fixed) vectors in the vector space V. Show that the set W of all linear combinations ab+bv 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.)
isuckatmath wrote: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?
isuckatmath wrote:Determine whether or not the given vectors in Rn form a basis for Rn.
v1=(3,-7,5,2), v2=(1,-1,3,4), v3=(7,11,3,13)
Since the vectors are in R4 and you have only three vectors (so clearly they cannot form a basis), try to find a vector in R4 which is not in the span of the set you've been given.
isuckatmath wrote:Let {v1,v2,...,vk} 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 v1,v2,...,vk,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?
