MA 405
April 12, 2009
Test 3  Study Guide
The topics for Test 3 are covered in Sections 6.1  6.3, 5.1  5.5 in the textbook. Learning objectives:
•
define and compute eigenvalues, eigenvectors, eigenspaces;
•
recognize diagonalizable matrices; be able to write a diagonalizable matrix as
A
=
PDP

1
, for some
invertible matrix
P
and diagonal matrix
D
; use it to compute powers of
A
.
•
apply the dot product, and use its properties; relationship between the dot product and matrix mul
tiplication;
•
identify orthogonal subspaces; define the orthogonal complement of a vector space;
•
define the four fundamental subspaces associated with a matrix; identify the relationship between
them; compute the basis for each; use their properties;
•
calculate the leastsquares solution of an inconsistent linear system; compute the residual and the
projection of a vector onto a subspace on
R
n
; calculate the equation of a line/parabola that best fits
a given set of data;
•
inner product spaces; definitions and properties; be able to check that an operation is an inner product;
•
identify orthogonal/orthonormal sets of vectors in an inner product space;
•
calculate the coordinates of a vector with respect to an orthogonal/orthonormal basis;
•
identify orthogonal matrices and use their properties;
1
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Practice Questions
1. Consider the matrix
A
=
2
3
0
4
3
0
0
0
6
.
(a) Find the eigenvalues of
A
.
(b) Find bases for the eigenspaces corresponding to each eigenvalue of
A
.
(c) Check that
A
is diagonalizable. Find the invertible matrix
P
that diagonalizes
A
and the diagonal
matrix
D
which is similar to A, that is,
P

1
AP
=
D
.
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 Spring '08
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 Linear Algebra, Orthogonal matrix, inner product, Inner product space

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