Condition (a) is not required. For example, the functions f(x)=x and g(x)=x2 are linearly independent.hgfalling said:Two or more equations are linearly independent if they are:
a) linear (ie, no terms with power > 1 and no multiplication of variables by each other),
b) independent (none of the equations can be derived algebraically from the others).
Linear independence refers to a set of vectors in a vector space that cannot be written as a linear combination of other vectors in the same space. In other words, none of the vectors in the set can be expressed as a combination of the others, making them all necessary for the span of the vector space.
To test for linear independence, you can use the determinant method or the rank method. The determinant method involves creating a matrix with the vectors as columns and calculating the determinant. If the determinant is non-zero, the vectors are linearly independent. The rank method involves creating a matrix with the vectors as rows and reducing it to row-echelon form. If all the rows are pivot rows, the vectors are linearly independent.
One example of linearly independent vectors is the standard basis vectors in a 3-dimensional space: (1, 0, 0), (0, 1, 0), and (0, 0, 1). Another example is the vectors (1, 2) and (3, 4) in a 2-dimensional space.
No, a set of linearly dependent vectors cannot be linearly independent. If a set of vectors is linearly dependent, it means that at least one vector can be written as a linear combination of the others, making it unnecessary for the span of the vector space. Therefore, the set cannot be linearly independent.
Linear independence is a fundamental concept in vector spaces. It allows us to determine if a set of vectors forms a basis for the vector space, which is a set of linearly independent vectors that can span the entire space. In other words, linear independence is a crucial property for creating a basis for a vector space.