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Linear Algebra Lecture 3 - Linear Independence and Bases Oskar Kędzierski 24 October 2022 DeĄnition Vectors v1,:::,vk P V are said to be linearly dependent if there exist real numbers α ,:::,α , not all of which are 0 such that 1 k α1v1 `:::`αkvk “ 0: Vectors v1,:::,vk P V are said to be linearly independent if they are not linearly dependent. By deĄnition, vectors v ,:::,v are linearly independent if 1 k α1v1 `:::`α v “0 implies that α1 “ ::: “ α “ 0. k k k Linear independence does not depend on the order of vectors hence we may talk about independent (Ąnite) sets. We assume that empty set is linearly independent. Linearly (In)dependent Vectors Let V be a vector space. Vectors v1,:::,vk P V are said to be linearly independent if they are not linearly dependent. By deĄnition, vectors v ,:::,v are linearly independent if 1 k α1v1 `:::`α v “0 implies that α1 “ ::: “ α “ 0. k k k Linear independence does not depend on the order of vectors hence we may talk about independent (Ąnite) sets. We assume that empty set is linearly independent. Linearly (In)dependent Vectors Let V be a vector space. DeĄnition Vectors v1,:::,vk P V are said to be linearly dependent if there exist real numbers α1,:::,αk, not all of which are 0 such that α v `:::`α v “0: 1 1 k k By deĄnition, vectors v1,:::,vk are linearly independent if α1v1 `:::`α v “0 implies that α1 “ ::: “ α “ 0. k k k Linear independence does not depend on the order of vectors hence we may talk about independent (Ąnite) sets. We assume that empty set is linearly independent. Linearly (In)dependent Vectors Let V be a vector space. DeĄnition Vectors v1,:::,vk P V are said to be linearly dependent if there exist real numbers α1,:::,αk, not all of which are 0 such that α v `:::`α v “0: 1 1 k k Vectors v1,:::,vk P V are said to be linearly independent if they are not linearly dependent.
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