core practice · coding problem
Identify rank and column space
A = [[1,2,3],[2,4,6],[1,1,0]]. What is rank(A)? Describe the column space in one sentence. Can you solve Ax=b for b=(3,6,2)?
The mechanism
Rank counts independent directions the matrix reaches. Regression lives in the column space of X; residuals are orthogonal to that subspace.
Expected evidence
rank=2 (row2=2×row1). Column space is span{(1,2,1),(2,4,1)}. Ax=b consistent — b is in column space.Try this drill in the practice workspace →
Hints when you need them
- Row reduction reveals pivot columns.
- b must be a linear combo of columns.
Watch for these mistakes
- Confusing rank with row count
- OLS when X is rank-deficient
- Ignoring near-collinearity