INFORMATION FOR FINAL
Our final will be on Saturday, March 19th from 1:003:00pm, in Kleiber 3.
The final is
comprehensive
.
So you need to know everything covered this
quarter.
However, I will focus more on the part that has not been covered by
the previous two midterms. Roughly, 4550% of the final will be on material from
chapters 2025, and the rest will be on the material that has been tested before.
Here is a list of the most important concepts from chapters 2025. (Since I gave
the lists of important concepts for the previous material already, I won’t repeat
them here. So you should combine these lists together to study for the final.)
(1) Change of basis and linear transformation using different bases. Understand
definition of component vectors. Know how to find change of basis matrices
and understand what they do.
Understand the definition of associated
matrices with respect to bases (other than the standard bases). Know how
to find the associated matrix with respect to a new basis from the associated
matrix with respect to an old basis.
(2) Diagonalization. Know that an
n
×
n
matrix is diagonalizable if and only
if it has
n
linearly independent eigenvectors, and know how to use this to
determine whether a matrix is diagonalizable. Know how to express
M
as
SDS

1
if
M
is diagonalizable. Know the connection between diagonalizing
M
and describing linear transformations with different bases.
(3) Orthogonal bases, orthonormal bases and GramSchmidt. Know the defi
nitions of orthogonal bases and orthonormal bases. Know how to use the
GramSchmidt process to find an orthogonal/orthonormal basis for a given
vector space.
(4) Diagonalizing symmetric matrices and orthogonal matrices. Know that a
real symmetric matrix is always diagonalizable. Know the results related
to the eigenvalues and eigenvectors of real symmetric matrices. Know the
definition of orthogonal matrices. Know how to express a real symmetric
matrix as
PDP

1
where
P
is an orthogonal matrix.
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 Spring '08
 chuchel
 Linear Algebra, Algebra, linear transformation

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