Linear Algebra Fundamentals

Question 1

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

Accepted Answer

c1

Answer

$(4,1)$

Answer

$(3,1)$

Answer

$(-2,3)$

Answer

$(2,1)$

Question 2

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

Accepted Answer

c3

Answer

$\begin{pmatrix}-1 & 0\0 & -1\end{pmatrix}$

Answer

$\begin{pmatrix}0 & 1\-1 & 0\end{pmatrix}$

Answer

$\begin{pmatrix}0 & -1\1 & 0\end{pmatrix}$

Answer

$\begin{pmatrix}1 & 0\0 & -1\end{pmatrix}$

Question 3

A $2	imes2$ matrix has determinant $-2$. Geometrically, what does this tell you about the linear transformation?

Accepted Answer

c1

Answer

It reverses orientation and scales area by $2$

Answer

It preserves orientation and scales area by $-2$

Answer

It reverses orientation and scales area by $-2$

Answer

It preserves orientation and scales area by $2$

Question 4

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

Accepted Answer

c3

Answer

Every vector is an eigenvector for some eigenvalue

Answer

Eigenvalues must always be positive real numbers

Answer

An eigenvector corresponding to eigenvalue $\lambda$ satisfies $A\mathbf{v}=\lambda\mathbf{v}$

Answer

Eigenvectors exist only for diagonal matrices

Question 5

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

Accepted Answer

c2

Answer

$A$ is orthogonal and preserves lengths in the $P$ basis

Answer

In the basis of columns of $P$, $A$ scales along coordinate axes by entries of $D$

Answer

$A$ sends every vector to an eigenvector of $P$

Answer

$A$ acts as a rotation in the basis given by columns of $P$

Question 6

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

Accepted Answer

c3

Answer

$P^2=0$ and $P^T=-P$

Answer

$P^2=P$ and $P^T=-P$

Answer

$P^2=P$ and $P^T=P$

Answer

$P^2=I$ and $P^T=P^{-1}$

Question 7

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

Accepted Answer

True

Answer

False

Question 8

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

Accepted Answer

True

Answer

False

Question 9

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

Accepted Answer

True

Answer

False

Question 10

Explain geometrically what multiplying a vector by a $2	imes2$ shear matrix does.

Accepted Answer

Multiplying by a $2	imes2$ shear matrix slides points parallel to one axis by an amount proportional to their coordinate on the other axis, preserving area but changing angles.

Question 11

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

Accepted Answer

The rank equals the dimension of the image (column space); by the rank-nullity theorem, rank plus nullity (dimension of kernel) equals the number of columns.

Question 12

Describe how to determine if a $3	imes3$ matrix is diagonalizable and outline a geometric interpretation if it is.

Accepted Answer

Compute eigenvalues and for each eigenvalue find independent eigenvectors; the matrix is diagonalizable if there are three linearly independent eigenvectors. Geometrically, diagonalizability means there is a basis of directions each scaled by a factor (the eigenvalues), so the transformation stretches/compresses along independent axes.