Final answer:
To find a minimal spanning set for the subspace spanned by S, we remove vectors that do not contribute to the span and ensure that remaining vectors are linearly independent. The minimal spanning set is {<0, -1>, <4, 4>} since these vectors are linearly independent and span the given subspace.
Step-by-step explanation:
To find a minimal spanning set for the subspace spanned by the set of vectors S={⠀⠀, <0>, <0,−1>, <4,4>, <0,0>} we need to identify which vectors add to the span without being linear combinations of each other. We can immediately remove the zero vector <0,0> from our set because it does not contribute to the span.
We look at the remaining vectors <0>, <0,−1>, and <4,4>. Vector <0> is already a multiple of <0,0> and can be removed. Now, consider <0, −1> and <4, 4>. Neither of these vectors can be written as a scalar multiple of the other, meaning they are linearly independent. Therefore, these two vectors form a minimal spanning set for our subspace, as any vector in the subspace can be written as a combination of these two vectors.
Step by Step Process:
Remove the zero vector <0,0> as it does not contribute to the span.
Remove any vector that is a scalar multiple of another vector in the set; in this case, <0>.
- Verify the remaining vectors are linearly independent.
- Conclude that <0, −1> and <4, 4> form the minimal spanning set