Sections, concepts, and problems
2.1. Linear transformations, null spaces, and ranges.
Structure-preserving maps between vector spaces (linear transformations), what
their null spaces (kernels) and ranges (images) are, what the words 'rank' and
'nullity' mean, and the Dimension Theorem.
2.2. The matrix representation of a linear transformation.
Coordinates of vectors relative to ordered bases, how to represent a linear
transformation as a matrix product in terms of ordered bases, the algebra of
linear transformations and how the set of all linear transformations V->W forms
a vector space.
2.3. Composition of linear transformations and matrix multiplication.
How L(V) is closed under composition (and in fact forms a ring), how to
associate a matrix to a linear transformation given ordered bases for the domain
and codomain, various properties of matrix multiplication, and some
applications.
2.4. Invertibility and isomorphisms.
Inverses and invertibility of linear transformations and matrices, isomorphisms
in the category of vector spaces and linear transformations, how L(V,W) is
isomorphic to the vector space of dim(W) x dim(V) matrices over F, and the
"standard representation" of vectors in a finite-dimensional vector space V.
2.5. The change of coordinate matrix. How changing coordinates from one
basis to another can be realized as multiplication by an appropriate matrix, how
to change the matrix of a linear transformation in terms of one basis into the
matrix in terms of another basis by conjugating by that same "change of basis"
matrix, and the notion of similar matrices.
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