Week 2 - Rank of a matrix, Determinants

Question 1

Rank of a matrix

Answer 1

1. The no. of NON-ZERO ROWS in the row-echelon form (REF)
2. The no. of leading ones
3. bounded by min{m, n} for matrix of size m x n

Question 2

How to write the solution space for Ax=0 if…
1. r=n
2. r<n

Answer 2

If r=n, Ax=0 has a unique solution
{x∈R^s | Ax=0} = {0}

If r<n, Ax=0 has infinitely many solutions
eg. {x∈R^s | Ax=0} = [sv1 + tv2 | s,t∈R}

Question 3

For A=m x n matrix, how many linearly independent (LI) vectors are there?

Answer 3

(n - r) LI vectors
where r is the rank

Question 4

Basis

Answer 4

A set of vectors B where every vector in the {solution} space is a UNIQUE LINEAR COMBINATION of the vectors in B

eg. {v1, v2}

Question 5

General solution to Ax=0

General solution to Ax=b
Note: Ap=b is a particular solution to Ax=b

^PRINCIPLE OF LINEARITY

Answer 5

To Ax=0, X=rv1 + sv2

To Ax=b, X =p + rv1 + sv2

1. If r=n, Ax=b has a unique solution
2. If r<n, Ax=b has infinitely many solutions

Question 6

How to calculate determinant using cofactor expansion along row i / down column j?

Answer 6

1. “a” - element in the matrix
2. “sign”
3. “minor” - determinant of the square (n-1)(n-1) matrix

Question 7

Effect of row operations on a matrix on the value of its determinant, for using UPPER-TRIANGULAR FORM (ie. all zeroes below main diagonal)

1. R2 -> kR2
2. R2 interchange R2
3. R2 -> R2 + R1
4. R2 -> R2 + kR1
5. R2 -> kR2 + R1

Answer 7

1. R2 -> kR2
   |B| = k|A|
2. R2 interchange R2
   |B| = -|A|
3. R2 -> R2 + R1
   |B| = |A| determinant is unchanged!
4. R2 -> R2 + kR1
   |B| = |A| determinant is unchanged!
5. R2 -> kR2 + R1
   |B| = k|A|

Question 8

“Sneaky trick” to find p

Answer 8

See ex.2

Question 9

Determinant of a matrix facts

Answer 9

1. If A has a row/column of zeroes, then |A|=0
2. If A has 2 identical rows/columns, then |A|=0

From ex.2 class,
|B| = 0 means…
1. LD
2. Infinitely many solutions = non-trivial
3. B is NOT invertible

|B| =/= 0 means…
1. LI
2. Only trivial solutions
3. B is invertible