Linear Algebra Fundamentals

Which of the following is a property of a vector in linear algebra?

What geometric transformation does a matrix represent when applied to a vector?

If $\lambda$ is an eigenvalue of matrix $A$ with eigenvector $\mathbf{v}$, which equation correctly describes this relationship?

How does the determinant of a matrix relate to the geometric interpretation of the corresponding linear transformation?

Which statement best describes an eigenvector of a linear transformation?

Given a matrix representing a linear transformation in $\mathbb{R}^2$, how can you determine if it includes a reflection component?

Every linear transformation in $\mathbb{R}^n$ can be represented by multiplying a vector by an $n \times n$ matrix.

An eigenvalue of zero means the linear transformation squashes the eigenvector onto the zero vector.

The sum of the eigenvalues of a matrix equals its determinant.

Explain how eigenvalues and eigenvectors relate to the geometric effect of a linear transformation represented by a matrix.

Match each linear algebra concept with its geometric interpretation: