Two arrows [1, 2] and [2, −1] added tip-to-tail, landing at (3, 1)
The home tip. You will see (3, 1) again as a mix of columns, as two lines, and as a solve.

The path

Vectors → pictures of Ax = b → projection → least squares → maps, eigen, SVD.

Lessons 1–14 are ready. The course stops here.

Ready

Vectors

A list is also an arrow. Add two, and the tip is a real point.

  1. Vectors as Lists and Arrows
  2. Dot Product and Magnitude
  3. Spanning Sets and Basis

Ready

Pictures of Ax = b

Those arrows become columns, then two lines. Elimination, the inverse, and the nullspace solve the same A.

  1. Ax as a Mix of Columns
  2. Row Picture of Ax = b
  3. Elimination and Back Substitution
  4. Inverse of A
  5. Column Space and Nullspace

Ready

Projection

The closest point in C(A) is a drop-perpendicular. When b is not in the columns you kept, that leftover is the fit.

  1. Projection
  2. Least Squares

Ready

Maps, eigen, SVD

The same product as a map of the plane. Then the arrows that only stretch. Then the circle that becomes an ellipse. Then those three maps multiply.

  1. Linear Maps
  2. Eigenvalues
  3. SVD, First Picture
  4. SVD Product