Part 8 : Linear Independence, Rank of Matrix, and Span

Assuming A = {a1, a2, a3, …., an} be a set of “n” number of vectors

and c1, c2, c3,..., cn be “n” number of constants (scalars which can have different values).

Linear combination of a set of vectors is formed when each vector in the set is multiplied with a scalar and the products are added together.

Then the linear combination of these vectors (and scalars) will be :

Linear combination

Let this linear combination be equal to 0.

This equation will be satisfied when all the scalars (c1, c2, c3, …, cn) are equal to 0.

But, if 0 is the only possible value of scalars for which the equation is satisfied then that set of vectors is called linearly independent.

A = {a1, a2, a3, …., an} is a set of linearly independent vectors only when for no value (other than 0) of scalars(c1, c2, c3…cn), linear combination of vectors is equal to 0.

For a 3x3 matrix, such as A

To find if rows of matrix are linearly independent, we have to check if none of the row vectors (rows represented as individual vectors) is linear combination of other row vectors.

Row vectors of matrix A

Turns out vector a3 is a linear combination of vector a1 and a2.

So, matrix A is not linearly independent. But, row vector a1 and a2 are linearly independent among each other.

Span

Set of linear combinations of some vectors having same number of components (or elements) is called span.

Some examples of linear combination of vectors a, b and c.

Assuming a span of three vectors a, b and c

Right arrow conversion could also be used to denote a vector

Hence, span is a set of all linear combinations of a, b and c.

This span also contains vectors a, b and c as they can also be represented as a linear combination.

Rank of Matrix

Maximum number of linearly independent rows in a matrix (or linearly independent columns) is called Rank of that matrix.

For matrix A, rank is 2 (row vector a1 and a2 are linearly independent).

Rank of matrix A is equal to rank of transpose of matrix A.

Row Equivalence

Matrix A is row equivalent to Matrix B if Matrix A could be obtained from Matrix B, by performing finite number of eliminations (as in Gauss-Jordan Elimination).

Taking example of two row equivalent matrices A and B.

Both of the matrices look similar

Performing these operations on matrix A

Matrix A after performing operations

We can see that matrix A could be obtained from matrix B by performing elimination. Hence, both of the matrices are row equivalent.

Row equivalent matrices will have same rank.

Read Part 9 : Vector Spaces and Subspaces

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