Explore geometric vectors in parallelograms, show opposite vectors are equal and parallel with magnitude, and express points on a line segment as convex combinations of endpoints, including division by ratios.

Compute 3A-2B and test parallelism via scalar multiples. Then find C one third from A to B and the midpoint M as (A+B)/2.

Explore systems of linear equations by expressing vectors as linear combinations, solving a two-by-two system for coefficients, and using matrix and augmented matrix forms to study consistency and solutions.

master elementary row operations to solve linear systems by transforming augmented matrices, including interchanging equations, scaling, and adding multiples to reach solutions for x and y.

Reduce a matrix to echelon form or reduced form using elementary row operations. Interchange rows and eliminate entries to place pivots and obtain the reduced row echelon form.

Analyze how the number of solutions to linear systems depends on consistency, pivots, and rank; distinguish unique, infinite, or no solutions using coefficient and augmented matrices.

Explain homogeneous systems with zero constants, show solving via reducing the coefficient matrix to RREF, and identify trivial and infinite solutions, plus the link to associated homogeneous and particular solutions.

Explore the geometric interpretation of solving a non homogeneous linear system by expressing solutions as a particular solution plus a homogeneous solution, forming a line parallel to a direction vector.

Explore basis and dimension by identifying a spanning, linearly independent set, removing dependent vectors, and recognizing that every basis has the same number of vectors, which defines the dimension.

Explore basis and dimension through three examples, using standard bases e1, e2, e3, determining independence, span, and pivots, and linking rank, null space, and dimension.

Explore the triple scalar product of three vectors, expressed as a 3x3 determinant with the first row as the first vector's components, and observe sign changes under vector interchange.

Identify a line in R3 by a point and a direction vector, and express it in vector, parametric, and symmetric forms, with examples converting between forms.

Explore rules of planes in R3 by deriving normal vectors from vectors in the plane via the cross product, and assess coplanarity using triple scalar products and dot products.

Compute the closest distance from a point to a line via the cross product, and find the closest point by projection, line equation, or plane intersection.

Explore square matrices, covering symmetry and the transpose, diagonal matrices, scalar multiples of the identity, and upper and lower triangular forms.

Define matrix powers for square matrices and illustrate computing P(A)=2A^2 + A - 3I for a 2×2 matrix, yielding the 2×2 result [[7,11],[0,18]].

Explore elementary matrices derived from the identity by a single row operation and how E times A applies that operation, yielding U with B equals U A.

Shows that matrix a and b are equivalent iff b equals u times a for some product of elementary matrices, illustrated by reducing a and b to the same form.