Algebraic and Geometric Multiplicity Practice Quiz (Linear Algebra)

📚 Topic Summary

In linear algebra, algebraic and geometric multiplicity describe properties of eigenvalues. The algebraic multiplicity of an eigenvalue is the number of times it appears as a root of the characteristic polynomial. The geometric multiplicity is the dimension of the eigenspace associated with that eigenvalue, which is the number of linearly independent eigenvectors corresponding to the eigenvalue. Understanding both multiplicities helps analyze the diagonalizability of a matrix.

🧮 Part A: Vocabulary

Match the term with its definition:

1. Algebraic Multiplicity
2. Geometric Multiplicity
3. Eigenvalue
4. Eigenspace
5. Characteristic Polynomial

Definitions:

1. The set of all eigenvectors corresponding to a specific eigenvalue, along with the zero vector.
2. A scalar $\lambda$ such that for a matrix $A$, $Av = \lambda v$ for some non-zero vector $v$.
3. The dimension of the eigenspace associated with an eigenvalue.
4. The number of times an eigenvalue appears as a root of the characteristic polynomial.
5. A polynomial whose roots are the eigenvalues of a matrix.

Match the terms to their correct definitions:

✍️ Part B: Fill in the Blanks

The algebraic multiplicity of an eigenvalue is found by examining the ______ ______. The geometric multiplicity, on the other hand, relates to the ______ associated with that eigenvalue. The geometric multiplicity is always ______ than or equal to the algebraic multiplicity, but it can never be ______. If the algebraic and geometric multiplicities are equal for all eigenvalues, then the matrix is ______.

🤔 Part C: Critical Thinking

Explain in your own words why the geometric multiplicity of an eigenvalue can never be greater than its algebraic multiplicity. Give an example to illustrate your explanation.