Linear Algebra · Advanced · 5 min
Column space
The column space is the set of all outputs a matrix can produce.
WHAT HAPPENS IF…Can two independent columns span all of three-dimensional space?
Loading the mathematical model…
Every output has coordinates (α,β,α+β), so the image is the plane z=x+y. Its normal is (−1,−1,1). Subtracting the target’s projection onto this normal gives the nearest point in the column space and a residual perpendicular to both columns.
A detail that matters
The column space lives in the output space, while the null space lives in the input space.
Keep the connection going
More linear algebra ↗