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CONTENT SUMMARY: pg 1.Lesson about nullity and rank of a linear transformation; kernel and image of linear transformation; general linear transformations; Example (mxn is 3x4)
How to visualize in higher dimensions; shift from affine space to vector space; points/vectors
4-dimensional space (algebraically)
4-dimensional space (geometrically)
linear transformation from 4dim to 3dim; Kernel and image of transformation as fundamental; nullity as dimension of the kernel; rank as dimension of the image
Definition of kernel vector; kernel property
Finding vectors with the kernel property for a transformation using row reduction
Definition of image vector; image property; pg 9: @ at least the columns of the transformation matrix have this image property
Finding vectors with the image property for a transformation using row reduction
Another approach to the image of a transformation; the column space of a matrix
The whole picture; kernel,image, nullity, rank
Important observations
relationship between the nullity and the rank; Rank-Nullity theorem
example: Linear transformation from 3dim space to 4dim space; kernel, image, rank, nullity
example continued; kernel
example continued; image; remark on relationship of columns in row reduction
example summary
exercises 17.(1:2); kernel property, image property
exercise 17.3 ; describe ker() and im(); (THANKS to EmptySpaceEnterprise)