Topics:

• Inverses Using Cofactors

• Cramer's Rule

• Vector Spaces

• Subspaces of Vector Spaces

• Span of a Subspace

• Linear Independence

• Basis of a Vector Space

• Dimension of a Vector Space

• Null Space of a Vector Space (titled Homogeneous Systems)

• Nullity of a Vector Space (titled Homogeneous Systems)

• Rank

• Transitions Between Basis (titled Coordinates)

• Magnitudes of Vectors

• Dot Product

• Cross Product

• Inner Products

Notes:

After finishing this course, I discovered that there are 7 different concepts that were crucial in this section and in later sections. Side note, as the topics got more complicated, it really helped me to be able to remember the properties linearly independent vectors had. Not only did the problem become much simpler, but there are other equivalent statements that I could use.

Here is a list of the most important things/concepts in the 2nd 1/3 of the semester:

• Span

• Linear independence

• Null space/nullity

• Basis

• Column Rank

• Transition matrices between two different basis (neither of which is the canon basis)

• Dot product/inner product for vectors

Topics:

• Gram-Schmidt process

• Orthogonal complements

• Projection of vectors onto subspaces

• Linear transformations

• Standard matrix representation of linear transformations

• Matrix representation of linear transformations with respect to 2 basis

• Kernel of a linear transformation

• Range of a linear transformation

• Eigenvalues, eigenvectors, and eigenspaces

• Diagonalization

• Orthogonal diagonalization

Notes:

From all of the problems I had done on these topics, I realized that the last section relied heavily on prior sections. I had to frequently do/use the following concepts:

• find the null space of A

• find the column space of A

• know the definition of being orthogonal and of being a normal vector

• use the inversion method on 2x2, 3x3, and 4x4 matrices

• find transition matrices between two basis

• calculate the determinant of 2x2, 3x3, and 4x4 matrices

Especially for the last 3 chapters, being able to find the determinant of 3x3 and 4x4 matrices quickly is a must! These last chapters were the hardest, in terms of grasping the concept. However, after a bunch of practice problems, they become much, much, MUCH easier.