Section 1.3: Subspaces

Annotations:

• Subspace: A subset, W, of a vector space, V, with the operations of addition and scalar multiplication defined on V. In addition the following conditions must hold: 1. [Closed under +] - For all x,y in W, x+y must be in W 2. [Closed under scalar] - For all c in F and x in W, cx must be in W 3. [Zero vector] - W must contain the zero vector 4. [Inverses] - For all x in W, there must exist a y in W s.t. x+y = 0

1. Theorem 1.3 (Conditions for a Subspace)

   Annotations:

   • Theorem 1.3 [Conditions for a Subspace] - Let V be a vector space and W be a subset of V. Then W is a subspace of V iff the following conditions hold: a) 0 exists in W b) For all x,y in W, x+y exists in W c) For all c in F and x in W, cx exists in W
2. Theorem 1.4 (Intersection of Subspaces)

   Annotations:

   • Theorem 1.4 [Intersection of Subspaces] - Any intersection of subspaces of a vector space V is a subspace of V.
3. Zero Subspace

   Annotations:

   • {0} is the zero subspace of a vector space V
4. Symmetric Matrix

   Annotations:

   • Symmetric Matrix: A matrix A s.t. A = A-transpose
5. Diagonal Matrix

   Annotations:

   • Diagonal Matrix: An nxn matrix M s.t. Mij = 0 whenever i does not equal j
6. Upper Triangular Matrix

   Annotations:

   • Upper Triangular Matrix: A matrix A s.t. Aij = 0 whenever i>j
7. Direct Sum

   Annotations:

   • Direct Sum: A vector space V is the direct sum of two subspaces W1 and W2 if the following: 1. W1 intersect W2 = {0} 2. W1 + W2 = V
8. Sum

   Annotations:

   • Sum: The sum of two nonemtpy subsets of a vector space V, S1 and S2, is defined as the following: S1+S2 = {x+y: x in S1 and y in S2}
9. Skew-Symmetric Matrix

   Annotations:

   • Skew-Symmetric Matrix: A nxn matrix M s.t. M-transpose = -M