Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text.(a) Geometrically, if λ is an eigenvalue of a matrix A and x is an eigenvector of A corresponding to λ, then multiplying x by A produces a vector λx parallel to x.(b) If A is an n × n matrix with an eigenvalue λ, then the set of all eigenvectors of λ is a subspace of Rn.
Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text.
(a) Geometrically, if λ is an eigenvalue of a matrix A and x is an eigenvector of A corresponding to λ, then multiplying x by A produces a
(b) If A is an n × n matrix with an eigenvalue λ, then the set of all eigenvectors of λ is a subspace of Rn.
A set S is said to be a subspace of a vector space V if the following conditions hold:
1. The zero vector belongs to S.
2. If then, .
3. If and c is a scalar then, .
We know that is a vector space of n-tuples of the form where
the elements belong to the set of real numbers .
Part a
We know that a number is said to be an eigenvalue of a matrix A if there is a non-zero vector x such
that .
It is given that is an eigenvalue of a matrix A and x is the corresponding eigenvector. Hence, .
We know that if we multiply a scalar c by a vector v then, the resulting vector cv is always parallel to the
vector v. Hence, is parallel to x.
Therefore, the given statement is correct.
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