Chapter 1.3 Definitions (Subspaces)

All definitions in Chapter 1.3

Subspace

A subset W of a vector space V over a field F is called a ________ of V if W is a vector space over F with the operations of addition and scalar multiplication defined on V.

What to Check for a Subspace W

Closure under addition, scalar multiplication, if W has 0 vector. All other VS properties automatically hold.

Theorem 1.3

Let V be a vector space and W a subset of V. Then W is a subspace of V if and only if the following conditions hold:

a) 0 is in W

b) x+y is in W when x,y are in W

c) cx is in W whenever c is in F and x is in W

Transpose

The _________ of an mxn matrix is the nxm matrix obtained from A after interchanging the rows with the columns; (At)ij = Aji

Symmetric Matrix

A matrix A such that A^t=A

Upper Triangular Matrix

An mxn matrix is called an _____ ________ ______ if all its entries lying below its diagonal entries are 0; that is Aij=0, when i>j.

Diagonal Matrix

An nxn matrix M is called a ________ ______ if Mij=0, whenever i≠j. That is, if all its non diagonal entries are 0.

Trace

The _____ of an nxn matrix M, denoted tr(M), is the sum of the diagonal entries of M; that is,

tr(M) = M11 + M22 + … + Mnn

Theorem 1.4

Any intersection of subspaces of a vector space V is a subspace of V.