1. 4.3 Subspaces of Vector Spaces
MAT 2401 Linear Algebra
4.3 Subspaces of Vector Spaces

2. HW
WebAssign 4.3
Written Homework

3. Preview
Continue to examine the structure of vector spaces.

4. Recall: Vector Spaces

5. Recall: Vector Spaces

6. Recall: Vector Spaces

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

8. 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.

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

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

11. 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!

12. 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?

13. Subspace

14. Subspace
May be we need only 2…

15. 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.

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

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

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

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

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

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