$\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{{\not{\mid}}} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&}$

## Section3.1Portfolio 3

Portfolio 3: Coordinates and Bases is due to Blackboard by 11:59 p.m. Monday, July 16th.

### Subsection3.1.1Purpose and Learning Standards

In the third portfolio you'll wrap up your study of the matrix-vector equation $A{\bf x} = {\bf b}$ by investigating how it can be used to determine, and translate among, coordinate systems on a vector space. This culminates in the centerpiece of the course, the Fundamental Theorem of Linear Algebra, which gives us a (first) way of tailor-making a linear transformation that has whatever effects we desire.

The learning standards included in this portfolio are:

• 3.A Determine the coordinate expression of a given vector with respect to a given basis.
• 3.B Construct and use invertible matrices that effect changes of coordinates.
• 3.C Given its effects on basis vectors, determine a matrix representing a given linear transformation.
• 3.D Calculate the determinant of a square matrix using a variety of strategies including row-reduction and LU factorization, and cofactor expansion.
• 3.X Demonstrate theoretical connections relating determinants of square matrices with the properties of those matrices.

### Subsection3.1.2Skills You'll Develop

In this portfolio you'll gain practice with the following skills that will not only promote your success in linear algebra, but grow as practitioners of pure and applied mathematics.

1. Understand how $\mathbb{R}^n$ is not only a vector space in itself, but also a model for all real vector spaces of its dimension.
2. Use a basis for a vector space to determine an associated coordinate system, and vice versa.
3. Use invertible matrices to change bases in a vector space.
4. "Custom-make" a matrix representing a given linear transformation, given choices of basis for its domain and codomain.
5. Use change-of-basis factorization to re-interpret a given linear transformation's matrix with respect to a new basis.
6. Calculate determinants of square matrices using efficient strategies, and discuss the geometric interpretation of determinants.
7. Use determinants and Cramer's Rule to solve square linear systems of equations.
8. Write clear, careful proofs of elementary mathematical facts that connect two or more related properties of a matrix (or vector space or linear transformation).

### Subsection3.1.3Knowledge You'll Gain

In this portfolio you'll gain familiarity with the following important content knowledge of linear algebra.

1. Coordinate vectors
2. Change-of-basis matrix
3. Fundamental Theorem, Part II: Linear transformation plus basis gives matrix
4. Similarity transformation
5. $LU$ factorization
6. Determinants
7. Minors and cofactors
8. Cramer's Rule for solving a system using determinants

You'll demonstrate that you've fulfilled the purpose of this assignment by completing the following sets of tasks. Each type of task will generate an artifact that you can include in your portfolio.

2. Engage regularly with your classmates through reading discussions, Flipgrid assignments, Piazza discussions, and/or online gatherings.