Advanced~7h

Linear Algebra Drills

The procedures, until they are fast

MA201 teaches each of these once, against one example, and moves on. This is the volume — thirty-three procedures from a dot product to a condition number, on questions generated fresh that never run out. Each stays green only while your recent answers hold.

Start this project See the modules

Modules: 9
Drills: 33
Build steps: 0
Time: ~7h

Walk out able to

Reduce a matrix, read its rank, find both eigenvalues of a $2 \times 2$ and project onto a line — without stopping to rebuild the method first.

What you drill

Vectors and systems

Norms, dot products, spans, elimination, the three verdicts.

Matrices and determinants

Products, inverses, the plane maps, 2×2 and 3×3 by hand.

Spaces, rank, eigenvalues

Independence, coordinates, pivots, subspaces, eigenvalues.

Orthogonality and fitting

Projection, Gram–Schmidt, normal equations, singular values.

The pitch

What you practise, and what you leave with

Not an artifact — a set of reflexes. Thirty-three procedures that are each either current or not, and a rail that says which.

You will practise

Norms, dot products, spans, elimination
Matrix products, inverses, determinants
Rank, the four subspaces, eigenvalues
Projection, Gram–Schmidt, least squares

Afterwards you can

Reduce and read a rank off a matrix
Find both eigenvalues of a 2×2
Project, and name the residual
Read a condition number for what it costs

Modules

9 modules, 34 items

Lessons explain one idea. Drills repeat it until it sticks. Build steps make something that exists afterwards.

1
Vectors and their lengths
~42 min
Measure, dot and classify a pair of vectors without rebuilding the formula first.
- LessonWhat this project is for4 min
- DrillMeasure a vector three ways8 min
- DrillDot two vectors8 min
- DrillRead the angle off the sign11 min
- DrillName what they span11 min
2
Solve and read a system
~47 min
Eliminate, back-substitute, and call the verdict before solving anything.
- DrillSolve a 2×2 system14 min
- DrillClear the entry under a pivot8 min
- DrillBack-substitute upward14 min
- DrillOne, none, or infinitely many11 min
3
Matrices as maps
~41 min
Apply, compose and invert a 2×2, and recall the map each one performs.
- DrillApply a matrix to a vector8 min
- DrillMultiply two matrices8 min
- DrillName the transformation11 min
- DrillInvert a 2×214 min
4
Spaces and independence
~36 min
Test a subset, spot a redundant direction, and read coordinates off a basis.
- DrillRun the subspace test11 min
- DrillIndependent, or redundant11 min
- DrillFind the coordinates14 min
5
Rank and the four subspaces
~42 min
Count pivots, deduce the count you were not given, and place each subspace.
- DrillCount the pivots20 min
- DrillRank plus nullity11 min
- DrillPlace the subspace11 min
6
Determinants
~50 min
Evaluate at 2×2 and 3×3, and carry a determinant through an operation.
- DrillTake a 2×2 determinant8 min
- DrillTake a 3×3 determinant20 min
- DrillTrack a row operation11 min
- DrillUse a determinant identity11 min
7
Eigenvalues and eigenvectors
~50 min
Verify, find, move and judge the eigenvalues of a 2×2.
- DrillRead an eigenvalue off a vector14 min
- DrillFind both eigenvalues14 min
- DrillMove the eigenvalue11 min
- DrillWill it diagonalise11 min
8
Orthogonality and fitting
~59 min
Project, classify, strip an overlap, and form the normal equations.
- DrillProject onto a line14 min
- DrillOrthogonal or orthonormal11 min
- DrillTake one Gram–Schmidt step14 min
- DrillForm the normal equations20 min
9
Spectral structure
~39 min
Classify a form, size the singular values, and read the conditioning.
- DrillClassify a quadratic form14 min
- DrillFind the singular values14 min
- DrillRead the condition number11 min