Linear independence and orthogonality are two different concepts in linear algebra. Linear independence refers to a set of vectors that cannot be written as a linear combination of each other. Orthogonality, on the other hand, refers to vectors that are perpendicular to each other. Therefore, it is possible for vectors to be linearly independent but not orthogonal.
In order for vectors to be linearly independent, they must have different directions. However, for vectors to be orthogonal, they must have perpendicular directions. Therefore, it is possible for vectors to be linearly independent but not orthogonal if they have different, but not perpendicular, directions.
An example of vectors that are linearly independent but not orthogonal is the standard basis vectors in three-dimensional space: i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1). These vectors have different directions, but are not perpendicular to each other.
Understanding the difference between linear independence and orthogonality is important in linear algebra because these concepts have different implications and applications. Linear independence is important for solving systems of linear equations and determining the dimension of a vector space. Orthogonality, on the other hand, is important for vector projections and orthogonal transformations.
No, vectors cannot be both linearly independent and orthogonal. If two vectors are orthogonal, they must have a dot product of 0. However, for vectors to be linearly independent, their dot product must be non-zero. Therefore, if vectors are orthogonal, they cannot be linearly independent.