Final answer:
Nullity is best described as the measure of the number of linearly dependent columns in a matrix, reflecting the dimension of the matrix's null space. It is an essential concept in linear algebra that is related to the Rank-Nullity Theorem.
Step-by-step explanation:
The definition of nullity of a matrix is most accurately described as a measure of the number of linearly dependent columns (or rows) in the matrix. This means that option 2 is the correct answer: it is a measure of the number of linearly dependent columns in a matrix. To further understand the concept of nullity, we should recall that it's related to the dimension of the null space or kernel of a matrix, which consists of all the vectors that, when multiplied by the matrix, give the null vector as a result.
The null vector is a vector where all components are zero, thus having no length and no direction. It serves as the identity element for vector addition. In terms of a matrix, nullity is a crucial concept when analyzing the solutions of a system of linear equations represented by the matrix.
For a matrix A, the connection between the nullity and the number of columns of A can be understood through the Rank-Nullity Theorem, which states that the sum of the rank (number of linearly independent columns) and the nullity of the matrix equals the total number of columns of the matrix. This is integral to linear algebra and matrix theory.
The Test of Independence mentioned in the context is different and relates to statistics, specifically to the assessment of the independence of two categorical variables. The term 'degrees of freedom' in a test of independence is calculated by the formula (number of columns-1)*(number of rows-1).