Summary and questions for repetition
I was planning to cover the Moore-Penrose inverse which allows one to solve the equation for any
(not necessarily square). Now my feeling is that it would be too much for a standard linear algebra course. This is the most easily accessible sourse.
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Give the definition and example of an orthonormal system. Prove that elements of such a system are linearly independent.
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"To know how a matrix acts on vectors, it is enough to know how it acts on the elements of an orthonormal basis." Explain.
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How can you reveal the elements of a matrix
from
?
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A linear mapping from one Euclidean space to another generates a matrix. Prove.
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Prove that an inverse of a linear mapping is linear.
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What is a linear mapping in the one-dimensional case?
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In case of a square matrix, what are the four equivalent conditions for the equation
to be good (uniquely solvable for all
)?
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Give two equivalent definitions of linear independence.
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List the simple facts about linear dependence that students need to learn first.
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Prove the criterion of linear independence.
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Let the vectors
be linearly dependent and consider the regression model
Show that here the coefficients
cannot be uniquely determined (this is a purely algebraic fact, you don't need to know anything about multiple regression).
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Define a basis. Prove that if
is a basis and
is decomposed as
then the coefficients
are unique. Prove further that they are linear functions of
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Prove that the terms in the orthogonal sum of two subspaces have intersection zero.
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Prove dimension additivity.
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Prove that a matrix and its transpose have the same rank.
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Prove the rank-nullity theorem.
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Prove the upper bound on the matrix rank in terms of the matrix dimensions.
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Vectors are linearly independent if and only if one of them can be expressed as a linear combination of the others.
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What can be said about linear (in)dependence if some vectors are added to or removed from the system of vectors?
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Prove that if the number of vectors in a system is larger than the space dimension, then such a system is linearly dependent.
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Give a list of all properties of rank that you've learned so far.
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Prove the inequality
and use it to give an alternative proof of the rank-nullity theorem.
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Prove that the definition of the space dimension does not depend on the basis used.
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