Engineering Analysis/Linear Independence and Basis

Linear Independence


A set of vectors   are said to be linearly dependent on one another if any vector v from the set can be constructed from a linear combination of the other vectors in the set. Given the following linear equation:


The set of vectors V is linearly independent only if all the a coefficients are zero. If we combine the v vectors together into a single row vector:


And we combine all the a coefficients into a single column vector:


We have the following linear equation:


We can show that this equation can only be satisifed for  , the matrix   must be invertable:


Remember that for the matrix to be invertable, the determinate must be non-zero.

Non-Square Matrix V


If the matrix   is not square, then the determinate can not be taken, and therefore the matrix is not invertable. To solve this problem, we can premultiply by the transpose matrix:


And then the square matrix   must be invertable:




The rank of a matrix is the largest number of linearly independent rows or columns in the matrix.

To determine the Rank, typically the matrix is reduced to row-echelon form. From the reduced form, the number of non-zero rows, or the number of non-zero columns (whichever is smaller) is the rank of the matrix.

If we multiply two matrices A and B, and the result is C:


Then the rank of C is the minimum value between the ranks A and B:




A Span of a set of vectors V is the set of all vectors that can be created by a linear combination of the vectors.



A basis is a set of linearly-independent vectors that span the entire vector space.

Basis Expansion


If we have a vector  , and V has basis vectors  , by definition, we can write y in terms of a linear combination of the basis vectors:




If   is invertable, the answer is apparent, but if   is not invertable, then we can perform the following technique:


And we call the quantity   the left-pseudoinverse of  .

Change of Basis


Frequently, it is useful to change the basis vectors to a different set of vectors that span the set, but have different properties. If we have a space V, with basis vectors   and a vector in V called x, we can use the new basis vectors   to represent x:




If V is invertable, then the solution to this problem is simple.

Grahm-Schmidt Orthogonalization


If we have a set of basis vectors that are not orthogonal, we can use a process known as orthogonalization to produce a new set of basis vectors for the same space that are orthogonal:

Find the new basis  
Such that  

We can define the vectors as follows:


Notice that the vectors produced by this technique are orthogonal to each other, but they are not necessarily orthonormal. To make the w vectors orthonormal, you must divide each one by its norm:


Reciprocal Basis


A Reciprocal basis is a special type of basis that is related to the original basis. The reciprocal basis   can be defined as: