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Cantspel
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We were going over linear independents in class and my professor said that if y1 and y2 are linearly independent then the ratio of y2/y1 is not a constant, but he never explained why it is not a constant.
Minor point -- you were going over linear independence in class. Linear independence is an attribute of a set of vectors of other elements that belong to a vector space.Cantspel said:We were going over linear independents in class
Linear independence refers to a set of vectors in a vector space that cannot be expressed as a linear combination of other vectors in the same space. In other words, no vector in the set can be written as a linear combination of the other vectors in the set.
To determine if a set of vectors is linearly independent, we can use the following criteria: if the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 is c1 = c2 = ... = cn = 0, then the set is linearly independent.
Linear independence is important in linear algebra because it allows us to solve systems of equations and perform other calculations more efficiently. It also helps us to understand the properties of vector spaces and their dimensions.
Yes, it is possible to reduce a set of linearly dependent vectors to a set of linearly independent vectors by performing operations such as row reduction and finding a basis for the vector space. This process is known as finding a linearly independent spanning set.
Linear independence is closely related to linear transformations because it helps us to understand the effects of these transformations on vector spaces. In particular, linear independence plays a key role in determining whether a linear transformation is one-to-one or onto (or both), and in finding the inverse of a linear transformation.