Werg22 said:Please refresh my memory; if a finite set S is L.I., then does this imply the existence of a set T of the same size (i.e. |T| = |S|) so that the elements T are pairwise orthogonal?
Linear independence is a mathematical concept that describes the relationship between vectors. It means that a set of vectors is considered linearly independent if none of the vectors can be written as a linear combination of the others.
To determine if a set of vectors is linearly independent, you can use the following criteria: If the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 is c1 = c2 = ... = cn = 0, then the vectors are linearly independent.
Orthogonality is a mathematical concept that describes the relationship between vectors. It means that two vectors are orthogonal if their dot product is equal to 0. In other words, the angle between the two vectors is 90 degrees.
To determine if two vectors are orthogonal, you can use the dot product formula: v1 · v2 = |v1| * |v2| * cosθ, where θ is the angle between the two vectors. If the dot product is equal to 0, then the vectors are orthogonal.
Linear independence and orthogonality are closely related concepts. In fact, if a set of vectors is both linearly independent and orthogonal, then it is also considered to be a basis for the vector space. This means that the vectors can be used to uniquely represent any vector in the space. Additionally, if a set of vectors is orthogonal but not linearly independent, it can be transformed into a linearly independent set by normalizing the vectors (making them unit vectors).