Linear Algebra Fundamentals

Which of the following is a requirement for two vectors to be orthogonal in $\mathbb{R}^n$?

If $A$ is a $2\times2$ matrix with columns $\begin{bmatrix}1\\0\end{bmatrix}$ and $\begin{bmatrix}0\\1\end{bmatrix}$, what linear transformation does $A$ represent?

Which statement correctly relates eigenvectors and linear transformations $T$ represented by matrix $A$?

Compute the determinant and infer the geometric effect: $\det\begin{pmatrix}2 & 0 \\ 0 & -3\end{pmatrix}$.

Matrix $A$ has eigenvalues $\lambda_1=3$ and $\lambda_2=1$ with independent eigenvectors. What is the limit of $A^n / 3^n$ as $n\to\infty$?

Which condition guarantees a matrix $A$ is diagonalizable over $\mathbb{R}$?

Explain how the columns of a matrix $A$ relate to the linear transformation it defines and how that connects to the image (column space).

Match each matrix property with its geometric or spectral meaning: