I'm practicing my proof-writing and was hoping you could let me know if this proof looks good. I would like to know if the proof is incorrect, if there are parts that are overly wordy/complicated, or if I'm missing some element of a proof that is helpful to see, if not strictly necessary.
The Prompt (from here):
A singleton vector is a linear independent set if and only if it is not the zero vector.
My Proof:
Suppose a vector v1 is a linear independent set. Let v1 be the zero vector. Then t1v1=0 for all values of t, which contradicts the fact that {v1} is a linearly independent set, so v1 is not the zero vector.
Suppose v1 is not the zero vector. Then t1v1=0 only when t1 = 0, which means that {v1} is a linearly independent set.