Matrices

• Matrices in Sage. Matrices in Sage.
• Elementary Row Operations on Matrices Using Sage to perform elementary row operations.
• Augmented Matrices. Augmenting a matrix with a column vector.
• The determinant of a matrix. Taking the determinant of a matrix.
• The sum of two matrices. Calculating the sum of two matrices.
• The product of two matrices. Calculating the product of two matrices.
• The transpose of a matrix. Calculating the transpose of a matrix.
• Unitary Matrices. Testing to see if a matrix is unitary
• The rank of a matrix. Calculating the rank of a matrix.
• The RREF of a matrix. Computing the RREF of a matrix.
• Finding the Pivot Columns of a Matrix. Finding the pivot columns of a matrix.
• Finding the Free Columns of a Matrix. Finding the free columns of a matrix.
• Testing a Matrix for Singularity. Testing a given matrix to see if it is singular (noninvertible).
• The inverse of a matrix. Computing the inverse of a matrix.
• Constructing Identity Matrices. A special command to create an identity matrix.

Systems of Linear Equations

• Right and Left System Solving Finding solutions to the equations $A\mathbf{x} = \mathbf{b}$ and $\mathbf{x}A = \mathbf{b}$.
• The Kernel of a Matrix Finding the kernel (or null space) of a matrix.
• The Column Space of A Matrix Finding the column space of a matrix.
• The Gram Schmidt Process Using the Gram-Schmidt process to construct an orthogonal (or orthonormal) basis.

Vector Operations

• Vectors in Sage. Vectors in Sage.
• Linear Combinations of Vector Building Linear Combinations in Sage.
• Random Elements of Vector Sets Using Sage to fetch random elements of vector spaces.
• Vector Addition & Subtraction. Adding and subtracting vectors in Sage.
• Finding a Scalar Product Finding the scalar product of a scalar and vector.
• Dot Product of Two Vectors Finding the dot product of two vectors.
• Cross Product of Two Vectors Finding the cross product of two vectors.
• Norm of a Vector Finding the norm (length) of a vector.
• Angle Between Two Vectors Finding the angle between two vectors.

Vector Spaces

• Vector Spaces in Sage Vector Spaces in Sage.
• Bases of Vector Spaces Using Sage to find bases of vector spaces.
• Vector Spans Finding the span of a finite set of vectors.
• Testing Sets for Linear Independence Testing a subset of a vector space for linear independence.

Inner Products

• Dot Product of Two Vectors Finding the dot product of two vectors.
• Norm of a Vector Finding the norm (length) of a vector.