Bases and Subspaces Worksheet Advanced Linear Algebra, Spring 2013 The vector space for this worksheet is P2; the set of polynomials of degree less that or equal to 2. We are given a basis for this...

Bases and Subspaces Worksheet Advanced Linear Algebra, Spring 2013 The vector space for this worksheet is P2; the set of polynomials of degree less that or equal to 2. We are given a basis for this vector space S =  1 + t + t 2 ; 1 + t; 1 + t 2 and a subspace W = span  1 + 2t + 3t 2 ; 2 + t; 1 + t + t 2 : Additionally we are given v 2 P2; but in terms of its coordinates: [v]S = 2 4 1 0 1 3 5 : 1. Does v 2 W? 2. Find a basis for W 3. Extend the previous basis to a basis of P2; and call this basis S 0 : 4. Characterize w 2 W in terms of [w]S0 5. If we know [w]S ; how do we Önd [w]S0?

Bases and Subspaces Worksheet Advanced Linear Algebra, Spring 2013 The vector space for this worksheet is P2, the set of polynomials of degree less that or equal to 2. We are given a basis for this vector space S = { 1 + t+ t2, 1 + t, 1 + t2 } and a subspace W = span { 1 + 2t+ 3t2, 2 + t, 1 + t+ t2 } . Additionally we are given v ∈ P2, but in terms of its coordinates: [v]S = 10 1  . 1. Does v ∈W? 2. Find a basis for W 3. Extend the previous basis to a basis of P2, and call this basis S′. 4. Characterize w ∈W in terms of [w]S′ 5. If we know [w]S , how do we find [w]S′? 1