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 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 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 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.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Expert Solution
Step 1
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 .
Step 2
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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