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make sure when the question asks for a prove you write the whole proof. like example 2 part c or question 3 prove that it is a subspace and if it is not explain why or question 4 part a prove that b is a basis of v for each part I, ii, iii .

Math 225, Fall 2015 Assignment 1

1. Review.
Consider the Euclidean Vector Spaces V = R3 and V = R4 with the usual
addition and scalar multiplication
(3) Find all values of k for Which the vector (3, 0, -1) is not in Span{(2, 2, -1), (k,2, 1)}
(b) Find all values of k for Which (3, 0, -1) is in Span{(2, 2, -1), (k,2, 1)}
(c) Find all values of k for Which these vectors are linearly dependent:
{(1,0,-1,0), (O,1,1,1), (2,3,1,k).}
For those k, express one vector as a linear combination of the others.
2. Abstract Vector Space.
Consider the set V = R2 With the addition EB and scalar multiplication >l<
defined by
(U17U2) G9 (7117712) = (”“1 + 111 + 1W2 + 112 – 1)
IC>K (U1,’lt2) 2 (km +k- 1,]C’lt2 -]€+ 1).
(3) Find the 0-vector for these operations, that is, the additive identity.
(HINT: It is not (0, 0))
(b) Find the vector -(2, 5) for these operations (HINT: It is not (-2, -5).)
(c) Prove that (R2, EB, >l<) is a vector space, that is, show that all 10 axioms
are satisfied.
3. Subspaces.
For the given vector spaces V and the given subsets W of V, either prove
that they are subspaces, or explain Why they are not.
(a) V=P3, W: {p(x) =a+bx+cx2+dx3 6P3 :p(1) =0}
(b) V=R4, W: {(w,:1:,y,z) ER4zw+2x-4y+2 =0}
(c) V2R3, W: {(a+b, (9+0, a-c) la, b, 06R}

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