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  • Subspaces associated to a linear transformation

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  • The range/image of a linear transformation is a subspace.
  • The inverse image of a subspace under a linear transformation is a subspace.
  • Definition of kernel/null space of linear transformation
  • The kernel/null space of a linear transformation is a subspace
  • Definition of generalized kernel/null space of linear transformation
  • Any vector space is the direct sum of the generalized kernel and gneralized range of a linear transformation on that space.
  • The kernels of powers of a linear transformation form an ascending chain
  • The range spaces of powers of a linear transformation form a descending chain
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  1. Any vector space is the direct sum of the generalized kernel and gneralized range of a linear transformation on that space.

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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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  • Aspen Summer 2016 - Entanglement Matters#1213
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