1. 3.2 Bases and Linear Independence
Every year
Linear Independence Day
is Celebrated in the US
on July 4
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2. Independence
A set of vectors is said to be (Linearly) independent if no vector is a linear combination of the other vectors.

3. Problem 10
Determine if the vectors are linearly independent

4. Problem 10 Solution

5. Problem 16
Determine if the vectors are Linearly Independent or dependent.

6. Problem 16 solution

7. What method could we use to systematically check for independence?

8. What method could we use to systematically check for independence?
Write the vectors as the columns of a matrix.
Find rref(A)
The columns with leading ones are independent.
The vectors with out leading ones are dependent.

9. Subspace
A subset W in Rn is a subspace if it has the following 3 properties
W contains the zero Vector in Rn
W is closed under addition (of two vectors are in W then their sum is in W)
W is closed under scalar multiplication

10. What are all of the possible vector subspaces in R2?
What are all of the possible subspaces in R3?

11. What are all of the possible vector subspaces in R2?
The zero vector
Lines passing through the origin
All of R2
What are all of the possible subspaces in R3?
Planes passing through the origin
All of R3

12. Examples of vector Subspaces
A(x) is a linear transformation from Rm to Rn
Is ker(A) a subspace in Rm?
Is the image (A) a subspace in Rn?
Is the line y = x a subspace of R2
Is the line y = x + 1 a subspace of R2

13. Examples of vector Subspaces
Is the line y = x a subspace of R2 yes
Is the line y = x + 1 a subspace of R2 No
A(x) is a linear transformation from Rm to Rn
Is ker(A) a subspace in Rm? yes
Is the image (A) a subspace in Rn? yes

14. Basis
A set of vectors is a basis of a subspace if the vectors are independent and they span the subspace.

15. Problem 28
Find a basis of the image (column space)

16. Problem 28 solution

17. Example 4 Consider the matrix
Column 2 is a multiple of column 1 and therefore adds no new information about the column space (Image)
Column 4 is a sum of columns 1 and 3 therefore it also gives no new information about the column space (Image)

18. Example 4
The image of A can be spanned by 2 vectors but not by 1 alone. (using 3 vectors would be redundant)
Therefore say the column vectors from the first and third column of the matrix form a basis of the subspace

19. Homework p odd