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sana2476
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If au + Bv + yw = a'u +B'v+y'w and a=a', then u,v,w are linearly dependent
As a side note, it would be helpful for you to be consistent with the letters you use. How did you happen to pick a, B, and y?sana2476 said:If au + Bv + yw = a'u +B'v+y'w and a=a', then u,v,w are linearly dependent
Linear dependence in vectors refers to the condition where one vector can be expressed as a combination of other vectors in the same vector space. In other words, if one vector can be written as a linear combination of other vectors, then those vectors are considered to be linearly dependent.
A set of vectors is linearly dependent if at least one vector in the set can be expressed as a linear combination of the other vectors. This can be determined by setting up a system of equations and solving for the coefficients of the linear combination. If the system has infinitely many solutions or no solutions, then the vectors are linearly dependent.
Linearly dependent vectors do not add any new information or increase the dimensionality of a vector space. They also do not form a basis for the vector space, which means they cannot be used to represent all vectors in that space. Therefore, linearly dependent vectors are not as useful as linearly independent vectors in solving problems in linear algebra and other fields of science and mathematics.
Yes, three or more vectors can be linearly dependent. As long as one vector can be expressed as a linear combination of the other vectors, the set is considered to be linearly dependent. However, it is important to note that for a set of n vectors, if n is greater than the dimension of the vector space, then the vectors are always linearly dependent.
Linearly dependent vectors can be used in real-world applications such as data analysis and pattern recognition. In these fields, linearly dependent vectors can represent similar or related data points, which can help in identifying trends and making predictions. Additionally, linearly dependent vectors can also be used in computer graphics and animation to create realistic movements and effects.