Linear Algebra Fundamentals

If $\mathbf{u}=(1,2)$ and $\mathbf{v}=(3,-1)$, what is $\mathbf{u}+\mathbf{v}$?

Which matrix represents a $90^\circ$ counterclockwise rotation in $\mathbb{R}^2$?

A $2\times2$ matrix has determinant $-2$. Geometrically, what does this tell you about the linear transformation?

Which statement is true about eigenvectors and eigenvalues of matrix $A$?

If $A$ is diagonalizable as $A=PDP^{-1}$, which is a correct geometric interpretation?

Consider projection matrix $P$ onto a line spanned by unit vector $\mathbf{u}$ in $\mathbb{R}^n$. Which properties hold for $P$?

If two vectors in $\mathbb{R}^3$ are orthogonal, their dot product is zero.

If a linear transformation $T:\mathbb{R}^2\to\mathbb{R}^2$ has two distinct real eigenvalues, then $T$ is diagonalizable.

A nonzero vector in the nullspace of matrix $A$ is an eigenvector with eigenvalue $0$.

Explain geometrically what multiplying a vector by a $2\times2$ shear matrix does.

How does the rank of a matrix relate to the dimensions of its image and kernel?

Describe how to determine if a $3\times3$ matrix is diagonalizable and outline a geometric interpretation if it is.