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suppose v1,...,vk are nonzero vectors with the property that vi.vj=0 whenever i is not equal to j. Prove that {v1,...,vk} is linearly independent.
Linear independence refers to a set of vectors in a vector space that cannot be expressed as a linear combination of each other. In other words, no vector in the set can be written as a linear combination of the other vectors, making them independent of each other.
To prove linear independence, you must show that the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0, where c1, c2, ..., cn are constants and v1, v2, ..., vn are the vectors in the set, is when all the constants are equal to 0. This can be done using various methods such as Gaussian elimination, determinants, or the definition of linear independence.
Proving linear independence is important in various fields such as mathematics, physics, and engineering. It allows us to determine if a set of vectors can serve as a basis for a vector space, which is crucial in solving systems of linear equations and understanding the properties of vector spaces.
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 in the set can be expressed as a linear combination of the other vectors. This violates the definition of linear independence, which requires each vector to be independent of the others.
Linear independence refers to a set of vectors that cannot be written as a linear combination of each other, while orthogonality refers to a set of vectors that are perpendicular to each other. Linear independence is a necessary but not sufficient condition for orthogonality. In other words, a set of linearly independent vectors can be orthogonal, but a set of orthogonal vectors may not be linearly independent.