) For each set listed in 1-4 below: (a) State whether the set of vectors spans a
ID: 2961635 • Letter: #
Question
) For each set listed in 1-4 below:
(a) State whether the set of vectors spans all of R3 or a plane, or a line or a point.
(b) List 3 vectors in the span (unless the span only consists of one point). For each vector, state the coefficients (i.e., c1, c2, etc.) that show the vector is in the span.
(c) List 1 vector not in the span (unless the span consists of all of R2 or all of R3). Show the calculation that proves that your vector is not in the span.
(d) State whether the set is linearly independent or linearly dependent.
(e) If linearly dependent, show at least one (non-trivial) set of coefficients that will allow you to
Explanation / Answer
There are short cuts here to the answers for these three sets.But first here is how to do this kind of problem:
You need to see how many linearly independent vectors you have in each set. You do this by making the vectors in the set the rows of a martix and using Gaussian elimination to reduce the matrix to reduce row echelon form. The nonzero rows of the reduced row echelon form will be a set of linearly independent vectors.
The span of a linearly independent set that contains only one vector will be a line; 2 vector will be a plane and 3 all of R^3.
In these particular cases, there are short cuts to the answer:
First, note that in the 3rd set, v1 and v3 are the same so we only need to keep one to find the span of the set. So lets get rid of v3. The zero vector contributes nothing to the span (except the zero vector) so we dont' need to worry about it. That leaves v1 and v4. But those vectors are scalar multiples of each other and so have the same span. Thus you really only have one vector in terms of the span and the span of one vector is a line.
In set # 1, notice that v1+v3 = v2. Thus v2 contributes nothing more to the span so we can ignore it. Also, 2v1 + 3v2 = v4 . So we can delete v4. That only leaves v1 and v2 which form a linearly independent set of 2 vectors, so the span is a plane.
The short cut for set 3 is something you should know to look for yourself. In set #1 you should know to look for something simple like v1+v3=v2. That leaves only 3 vectors to do the Gaussian elimination on so it makes you work easier. (Realizing that v4 can be deleted only occurs if you're experienced doing these problems)
The answer for set 2 is that there are 3 linearly independent vectors (easiest to just do the Gaussian elimination) and 3 linearly independent vector in R3 must span all of R3.
So I'd say that for someone just learning this material for the first time, you should look for something real easy (like analyzing set 3) but then go ahead and do the Gaussian elimination.
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