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atakel
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Let t€R be fixed. Show that {u,v}CR^2 with u=(cost,sint), v=(-sint,cost) is a linearly inpedendent set.
Linear independence refers to a set of vectors that cannot be expressed as a linear combination of other vectors in the same vector space. In simpler terms, it means that none of the vectors in the set can be created by combining the others using scalar multiplication and addition.
To determine if a set of vectors is linearly independent, you can use the determinant method or the rank method. The determinant method involves creating a matrix with the vectors as columns and finding its determinant. If the determinant is non-zero, the vectors are linearly independent. The rank method involves creating an augmented matrix with the vectors and reducing it to row-echelon form. If the rank of the matrix is equal to the number of vectors, they are linearly independent.
Yes, a set of two vectors can be linearly independent as long as they are not scalar multiples of each other. This means that they cannot lie on the same line or be parallel to each other.
Linear independence is the opposite of linear dependence. If a set of vectors is linearly independent, it means that they are not linearly dependent, and vice versa. In other words, if a set of vectors is not linearly independent, it is linearly dependent, and can be expressed as a linear combination of other vectors in the same vector space.
Linear independence is an important concept in linear algebra because it allows us to determine the number of linearly independent equations in a system and find the solutions. It also helps in determining the basis of a vector space and solving problems involving vectors and matrices. Additionally, linear independence is crucial in many applications, such as data analysis, computer graphics, and machine learning.