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Can non orthogonal vectors be linearly independent?
i and i+j are linearly independent, but not orthogonal. For example, in R2, the vectors <1, 0> and <1, 1,> are independent since the only way to have a<1, 0>+ b<1, 1>= 0 is to have a= 0 and b= 0. But they are NOT “orthogonal”- the angle between them is 45 degrees, not 90.
Why say orthogonal instead of perpendicular?
And since all Right Angles are Equal (one of Euclid’s Axioms) every Orthogonal Line implies 4 Rays from the point of intersection. Notice that one can prove (probably, depending on who you are and which axioms you choose) that every Orthogonal is a Perpendicular, and every Perpendicular is an Orthogonal.
Why are orthogonal vectors linearly independent?
Orthogonal vectors are linearly independent. If we have n linear independent vectors in Rn, they automatically span the space because the fundamental theorem of linear algebra shows that the image has then dimension n. A vector w ∈ Rn is called orthogonal to a linear space V , if w is orthogonal to every vector v ∈ V .
How do you know if you are orthogonal?
We say that 2 vectors are orthogonal if they are perpendicular to each other. i.e. the dot product of the two vectors is zero. Definition.
What is the relationship between independent, correlation and orthogonality?
Unlike that independent is a stronger concept of uncorrelated, i.e., independent will lead to uncorrelated, (non-)orthogonal and (un)correlated can happen at the same time. I am being the TA of probability this semester, so I make a short video about Independence, Correlation, Orthogonality.
Is it necessary for an orthogonal process to be independent?
Therefore, it is not necessary for orthogonal processes to be independent. Independence in random processes means that if you have any foreknowledge about one process, you will not be able to have any conclusion about the other! However, this is not the case with orthogonal processes.
How is statistical independence related to uncorrelatedness?
Statistical independence means that the joint PDF of two random variables can be written as the product of the individual PDFs: Independence implies uncorrelatedness, but the opposite is generally not true. If X and Y are jointly Gaussian, then independence and uncorrelatedness are equivalent.
When are orthogonality and uncorrelatedness of a process are equivalent?
If X and Y are jointly Gaussian, then independence and uncorrelatedness are equivalent. Consequently, in the special case that X and Y are jointly Gaussian and at least one of them has a mean of zero, then orthogonality, uncorrelatedness, and independence are all equivalent.