Educational Cards
Learn from video content, text, and interactive tasks
Filters
Non-commutativity: order usually matters
Even when both AB and BA exist, they often differ. Different pipelines deform space differently...
Column picture of $AB$: $A$ times each column of $B$
Column j of AB equals A times column j of B. Thus the column space of AB lies inside the column...
Row picture: rows of $A$ dot columns of $B$
Entry (i,j) of AB is row i of A dotted with column j of B. Useful when interpreting rows as...
Powers and repeated transformations
Applying the same map k times corresponds to A^k when well-defined. Diagonalization later makes...
Looking ahead: encode all sequential computation pipelines
Numerical pipelines in graphics, robotics, and recurrent nets are long matrix chains. This chapter...
Coordinates depend on basis; geometry does not
Two observers can use different skewed grids while agreeing on the same arrows in space. Their...
Transformations in alternate coordinates sandwich
If M_textyours is the matrix of a linear map in your coordinates, the same geometric transformation...
Parsing the "backwards" feeling
Students often feel the change-of-basis story runs backward. The resolution is to separate geometry...
Trace and determinant stay similar
Similarity B^-1MB preserves trace and determinant even when individual entries scramble [@2]. That...
Worked mental checklist for change of basis
A reliable workflow: identify columns of B, confirm invertibility, then multiply or sandwich as the...
Lead-in: eigenbasis is the ultimate friendly coordinate change
When enough eigenvectors span the space, a matrix B built from them diagonalizes M = B D B^-1 with...
Eigenvector: stays on its span, only stretches or flips
An eigenvector mathbfv satisfies Amathbfv = lambda mathbfv: the map acts on mathbfv purely by the...