The vectors you can reach as Ax form the column space of A: all combinations of its columns. Vectors are linearly independent if none is a combination of the others. A basis is an independent set that spans a space, and the number of vectors in it is the dimension. The rank of A is the dimension of its column space, equal to the number of pivots in elimination and to the number of independent columns (and of independent rows).
rank(A)+dimnull(A)=n(1)
The null space of A is the set of x with Ax=0: the inputs the map sends to nothing. Its dimension, the nullity, is n−rank(A) (the rank-nullity theorem). This decides every linear system. Ax=b has a solution exactly when b is in the column space; when it does, the solutions are one particular solution plus anything in the null space, so the solution is unique exactly when the null space is {0}.
A matrix with a redundant rowTake A=121240361. Row 2 is twice row 1, so only two rows are independent: rank 2 and nullity 3−2=1. Solving Ax=0 gives x1=−x3 and x2=−x3, so the null space is spanned by (1,1,−1). Then Ax=b is solvable only when b2=2b1, and when it is solvable there are infinitely many solutions, a line of them.
Column space: every Ax; Ax=b is solvable exactly when b lies in it.
Null space: all x with Ax=0; if it holds more than 0, solutions are never unique.
Rank-nullity: rank plus nullity equals the number of columns.
Rank in data: redundant assetsIf one asset's return series is exactly a combination of others (an ETF and its constituents, or a sum of two sector returns), the data matrix loses rank, its covariance is singular and a regression on all of them has no unique solution. Real data is rarely exactly dependent but often nearly so, and that is the ill-conditioning of the previous topic.
Lab · what's graded
The rank15 pts
The nullity15 pts
A null vector15 pts
When is the system solvable15 pts
How many solutions20 pts
Redundant assets20 pts
0/ 6 answered
1Numeric answer15 ptsNot answered
Let A=121240361. What is the rank of A?
rank=accepted: ± 0.001
2Numeric answer15 ptsNot answered
A matrix has 4 rows, 6 columns and rank 3. What is the dimension of its null space?
nullity=accepted: ± 0.001
3Choice15 ptsNot answered
Let A=121240361. Which vector satisfies Ax=0?
4Choice15 ptsNot answered
Let A=121240361. For which right-hand side b does Ax=b have a solution?
5Match20 ptsNot answered
Match each situation to what happens to its solutions.
Pick the partner of each item on the left.
A square matrix of full rank
Rank below the number of columns, and b in the column space
b outside the column space
Full column rank and b=0
6Explain in words20 ptsNot answered
Two assets in a regression are almost duplicates (one is nearly twice the other). In two sentences, say why the regression becomes unstable.
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