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  • Linear (in)dependence

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  • Definition of linear dependence relation on a set of vectors
  • Definition of trivial linear dependence relation on a set of vectors
  • Determine if a particular set of vectors in R^3 in linearly independent
  • Definition of linearly independent set of vectors: if a linear combination is 0
  • Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors
  • Definition of linearly dependent set of vectors: one of the vectors can be written as a linear combination of the other vectors
  • Theorem: a set of vectors is linearly independent if and only if whenever a linear combination is 0
  • A set of vectors is linearly independent if and only if the homogeneous linear system corresponding to the matrix of column vectors has only the trivial solution.
  • A set of vectors is linearly independent if and only if the matrix of column vectors in reduced row-echelon form has every column as a pivot column.
  • If a set of vectors contains the 0 vector
  • A set of two vectors is linearly dependent if and only if neither is a scalar multiple of the other.
  • If a set of vectors in R^n (or C^n) contains more than n elements
  • A subset of a linearly independent set is linearly independent.
  • A set is linearly independent if and only if the set of coordinate vectors with respect to any basis is linearly independent.
  • Removing a linearly dependent vector from a set does not change the span of the set.
  • Adjoining an element not in the span of a linearly independent set gives another linearly independent set.
  • Any linearly independent set can be expanded to a basis for the (sub)space
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  1. Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors

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  • Example of a linear transformation on R^3: rotation#23892
  • Example of a sum of vectors interpreted geometrically in R^2#23796
  • Example of (echelon matrix/matrix in (row) echelon form)#23541
  • Example of finding the inverse of a 2-by-2 matrix by row reducing the augmented matrix#23528
  • Example of finding the inverse of a 2-by-2 matrix by using a formula#23529
  • Example of finding the inverse of a 3-by-3 matrix by row reducing the augmented matrix#23530
  • Example of finding the inverse of a 3-by-3 matrix by using Cramer's rule#23531
  • Example of linear combination of vectors in R^2#23801
  • Example of matrix-vector product#23485
  • Example of multiplying 2x2 matrices#23492
  • Example of multiplying 3x3 matrices#23493
  • Example of multiplying matrices#23495
  • Example of multiplying nonsquare matrices#23494
  • Example of putting a matrix in echelon form#23537
  • Example of putting a matrix in echelon form and identifying the pivot columns#23538
  • Example of row reducing a 3-by-3 matrix#23546
  • Example of row reducing a 4-by-4 matrix#23547
  • Example of solving a 3-by-3 homogeneous matrix equation#23766
  • Example of solving a 3-by-3 homogeneous system of linear equations by row-reducing the augmented matrix#23762
  • Example of solving a 3-by-3 matrix equation#23765
  • Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix#23761
  • Example of using the echelon form to determine if a linear system is consistent.#23772
  • Example of vector-scalar multiplication in R^2#23794
  • Example of writing a given vector in R^3 as a linear combination of given vectors#23803
  • Examples#23886
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