A basis must be linearly independent AND span the space. {(1,0),(0,1)} is the standard basis — 2 independent vectors that span ℝ². The first set has 3 vectors and is not linearly independent. The second set is linearly dependent (each vector is a scalar multiple of the other). The fourth contains the zero vector, which makes any set linearly dependent.
Question 2 True / False
Nearly every linearly independent set of vectors in ℝ³ is a basis for ℝ³.
TTrue
FFalse
Answer: False
A basis must satisfy two conditions: linear independence AND spanning. Two linearly independent vectors in ℝ³ span only a plane through the origin, not all of ℝ³. A basis for ℝ³ requires exactly 3 linearly independent vectors. Having too few independent vectors means the set cannot span the full space.
Question 3 Short Answer
If a vector space has dimension n, what is the maximum number of vectors in any linearly independent set?
Think about your answer, then reveal below.
Model answer: n
The dimension n is simultaneously the size of every basis, the maximum size of any linearly independent set, and the minimum size of any spanning set. Any linearly independent set with n vectors automatically spans the space and is therefore a basis. Trying to find n+1 linearly independent vectors always fails — the (n+1)th must be a linear combination of the others.