Two sequences x and y are said to span a space S over field F if ad only if "every sequence z in S can be expressed as z = ax +by for some a,b ∈ F "
A student thus wrote: "p and q span S because w∈S,0∈F and w = 0p + 0q" (Note that w∈S,0∈F and w = 0p + 0q are all correct)
Here comes the question:
1.Explain why the student's argument might be false/wrong?
2.Explain why the student's argument might be true/correct?
This is what I write:
For statement 1, it might be false because it only shows that sequence w can be expressed in terms of op+oq,where w=0 ,so it doesn't show that all the sequence w in S can be expressed in the form ?p+?q
For statement 2, it might be true because w=0, this is true by default.
the o above is zero (although it really seems like "o")
For statement 1, is my argument sufficient enough to support my opinion? and for statement 2, how can I elaborate more on the true by default concept?