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1 Some questions to ponder
- why is Axler even talking about polynomials in Chapter 2.A?
- Polynomials can also form a vector space, and thus the same rules apply. By talking about polynomials, Axler shows some of the unifying power of vector spaces.
- is there an intuitive way to describe the span of a set of
vectors?
- The span goes from the set of two non-collinear vectors on a plane to the set of every vector on the plane
- is there an easy or quick way to check if a set of vectors is
linearly independent?
- Represent as system of equations, then see if you can get to reduced row echelon form.
- what is the relationship between linear independence (of a set of
vectors) and systems of equations?
- Treating scalars as variables, you can use a system of equations to represent all possible linear combinations, convert that to a matrix, then use that to determine linear independence.
- what is the relationship between linear independence (of a set of
vectors) and nonsingularity (of a matrix)?
- If vectors are collinear when represented as the column of a matrix then the determinant will be 0. When vectors are collinear, they are not linearly independent. Therefore, when the determinant is 0, the vectors are linearly dependent.
- what is the relationship between linear independence (of a set of
vectors) and direct sum (of subspaces)?
- The product of a direct sum must be linearly independent because by definition all items in a subspace must be represented uniquely.