Understanding Linear Dependence and Rank in Linear Equations, Slides of Engineering Mathematics

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Rank

Rank

• In a system of linear equations, the notion of “rank” measures the number of essentially different equations.

They are multiple of each other. We can remove one of them without altering the solutions.

The representation using

augmented matrix

• Recall that the augmented matrix representation of a system of linear equations.

The third equation is the sum of the first and second equation

The third row is equal to the sum of the first and second rows

Recall: Linearly dependence

• Definition: m vectors v 1 , v 2 , …, v m (each of them have n components) are linearly dependent if there are m real numbers, a 1 , a 2 , …, am, not all zero, such that the linear combination is equal to zero a 1 v 1 + a 2 v 2 + … + am v m= 0

Theorem

v 1 , v 2 , …, v m are linearly dependent if and only if one of them is a linear combination of the others.

Proof () Suppose that v 1 , v 2 , …, v m are linearly dependent. By definition, we can find m real numbers, a 1 , a 2 , …, am, not all zero such that a 1 v 1 + a 2 v 2 + … + am v m= 0. Suppose that ai0. Then v i = (-1/ai)(a 1 v 1 + …+ai-1 v i-1+ ai+1 v i+1+ … + am v m)

The i-th vector v i is a linear combination of the others.

Theorem

v 1 , v 2 , …, v m are linearly dependent if and only if one of them is a linear combination of the others.

Proof () Suppose that v i is a linear combination of the other vector: v i = c 1 v 1 + …+ci-1 v i-1+ ci+1 v i+1+ … + cm v m

for some constants c 1 , …,ci-1, ci+1,…cm. Then 0 = c 1 v 1 + …+ci-1 v i-1– v i+ ci+1 v i+1+ … + cm v m

The i-th coefficient is non-zero (it is equal to -1). The zero vector can be expressed as a linear combination of v 1 , v 2 , …, v m, in which not all coefficients are zero.

Definition: Row-rank of a matrix

• Given a matrix M , the row-rank of M is the maximum number of linearly independent rows in M.
• Example:
  • Because the third row is the sum of the other two rows, i.e., is a linear combination of the other two rows., by the theorem in p.32, the three rows are linearly dependent.
  • The first and second row are linearly independent.
  • Therefore the maximum number of linearly independent rows is 2.
  •  the row-rank of this matrix is equal to 2.

Example

• What is the row-rank of?
• What is the row-rank of?

Ans: 2

Ans: 4