# Understanding Linear Algebra

## AppendixASage Reference

We have introduced a number of Sage commands throughout the text, and the most important ones are summarized here in a single place.
Accessing Sage
In addition to the Sage cellls included throughout the book, there are a number of ways to access Sage.
1. There is a freely available Sage cell at sagecell.sagemath.org.
2. You can save your Sage work by creating an account at cocalc.com and working in a Sage worksheet.
3. There is a page of Sage cells at gvsu.edu/s/0Ng. The results obtained from evaluating one cell are available in other cells on that page. However, you will lose any work once the page is reloaded.
Creating matrices
There are a couple of ways to create matrices. For instance, the matrix
\begin{equation*} \begin{bmatrix} -2 \amp 3 \amp 0 \amp 4 \\ 1 \amp -2 \amp 1 \amp -3 \\ 0 \amp 2 \amp 3 \amp 0 \\ \end{bmatrix} \end{equation*}
can be created in either of the two following ways.
1. matrix(3, 4, [-2, 3, 0, 4,
1,-2, 1,-3,
0, 2, 3, 0])

2. matrix([ [-2, 3, 0, 4],
[ 1,-2, 1,-3],
[ 0, 2, 3, 0] ])

Be aware that Sage can treat mathematically equivalent matrices in different ways depending on how they are entered. For instance, the matrix
matrix([ [1, 2],
[2, 1] ])

has integer entries while
matrix([ [1.0, 2.0],
[2.0, 1.0] ])

has floating point entries.
If you would like the entries to be considered as floating point numbers, you can include RDF in the definition of the matrix.
matrix(RDF, [ [1, 2],
[2, 1] ])

Special matrices
The $$4\times 4$$ identity matrix can be created with
identity_matrix(4)

A diagonal matrix can be created from a list of its diagonal entries. For instance,
diagonal_matrix([3,-4,2])

Reduced row echelon form
The reduced row echelon form of a matrix can be obtained using the rref() function. For instance,
A = matrix([ [1,2], [2,1] ])
A.rref()

Vectors
A vector is defined by listing its components.
v = vector([3,-1,2])

The + operator performs vector and matrix addition.
v = vector([2,1])
w = vector([-3,2])
print(v+w)

A = matrix([[2,-3],[1,2]])
B = matrix([[-4,1],[3,-1]])
print(A+B)

Multiplication
The * operator performs scalar multiplication of vectors and matrices.
v = vector([2,1])
print(3*v)
A = matrix([[2,1],[-3,2]])
print(3*A)

Similarly, the * is used for matrix-vector and matrix-matrix multiplication.
A = matrix([[2,-3],[1,2]])
v = vector([2,1])
print(A*v)
B = matrix([[-4,1],[3,-1]])
print(A*B)

Operations on vectors
1. The length of a vector v is found using v.norm().
2. The dot product of two vectors v and w is v*w.
Operations on matrices
1. The transpose of a matrix A is obtained using either A.transpose() or A.T.
2. The inverse of a matrix A is obtained using either A.inverse() or A^-1.
3. The determinant of A is A.det().
4. A basis for the null space $$\nul(A)$$ is found with A.right_kernel().
5. Pull out a column of A using, for instance, A.column(0), which returns the vector that is the first column of A.
6. The command A.matrix_from_columns([0,1,2]) returns the matrix formed by the first three columns of A.
Eigenvectors and eigenvalues
1. The eigenvalues of a matrix A can be found with A.eigenvalues(). The number of times that an eigenvalue appears in the list equals its multiplicity.
2. The eigenvectors of a matrix having rational entries can be found with A.eigenvectors_right().
3. If $$A$$ can be diagonalized as $$A=PDP^{-1}\text{,}$$ then
D, P = A.right_eigenmatrix()

provides the matrices D and P.
4. The characteristic polynomial of A is A.charpoly('x') and its factored form A.fcp('x').
Matrix factorizations
1. The $$LU$$ factorization of a matrix
P, L, U = A.LU()

gives matrices so that $$PA = LU\text{.}$$
2. A singular value decomposition is obtained with
U, Sigma, V = A.SVD()

It’s important to note that the matrix must be defined using RDF. For instance, A = matrix(RDF, 3,2,[1,0,-1,1,1,1]).
3. The $$QR$$ factorization of A is A.QR() provided that A is defined using RDF.