4.3 Subspaces of Vector Spaces MAT 2401 Linear Algebra 4.3 Subspaces of Vector Spaces http://myhome.spu.edu/lauw

HW WebAssign 4.3 Written Homework

Preview Continue to examine the structure of vector spaces.

Recall: Vector Spaces

Recall: Vector Spaces

Recall: Vector Spaces

Question? Suppose V is a vector space. If W is a subset of V, is W also a vector space?

Subspace A nonempty subset W of a vector space V is called a subspace of V if W is a vector space under the operations of addition and scalar multiplication defined in V.

Example 1 Subspace of R2? W1 = {(x,y)| x-2y=0}

Example 1 Subspace of R2? W1 = {(x,y)| x-2y=0} Let’s check the 10 axioms!

Hold on…check only 4 of them…. Example 1 Subspace of R2? W1 = {(x,y)| x-2y=0} Hold on…check only 4 of them…. Let’s check the 10 axioms!

Example 1 Subspace of R2? W1 = {(x,y)| x-2y=0} Hold on…check only 4 of them…. Is it because you have only 4 fingers?

Subspace

Subspace May be we need only 2…

Theorem If W is a nonempty subset W of a vector space V, then W is a subspace of V if and only if 1. If u and v are in W, then u+v is in W. 2. If u is in W and c is any scalar, then cu is in W.

Example 1 Subspace of R2? W1 = {(x,y)| x-2y=0}

Example 1(b) Subspace of R2? W2 = {(x,y)| x,y≥ 0}

Example 1(c) Subspace of R2? W3 = {(0,0)}

Example 2(a) Subspace of M2,2? S2 = the set of all 2x2 symmetric matrices

Example 2(b) Subspace of M2,2? W = the set of all 2x2 singular matrices

Theorem If V and W are both subspaces of a vector space U. Then the intersection, VW, is also a subspace of U.