Find the eigen value of [5]

Let an operation $*$ be defined on $\mathbb{Q}^+$ by $a * b = \frac{ab}{2}$. Then show that $\mathbb{Q}^+$ forms a group. [5]

Define system of linear equations. When a system of equation is consistent? Determine if the system is consistent:
[10]

Find the LU factorization of [10]

If $A = \begin{bmatrix} 7 & 2 \\ -4 & 1 \end{bmatrix}$, find a formula for $A^n$, where $A = PDP^{-1}$, $P = \begin{bmatrix} 1 & 1 \\ -1 & -2 \end{bmatrix}$ and $D = \begin{bmatrix} 5 & 0 \\ 0 & 3 \end{bmatrix}$ [5]

Find a least square solution of the inconsistent system $Ax = b$ for [10]

Define ring and show that set of real numbers with respect to addition and multiplication operation is a ring. [5]

Define linear transformation with an example. Let and define a transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ by $T(x) = Ax$ then a. find $T(v)$ b. find $x \in \mathbb{R}^2$ whose image under $T$ is $b$ [10+0]

Let $A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ and define $T: \mathbb{R}^2 \to \mathbb{R}^2$ by $T(x) = Ax$, find the image under $T$ of [5]

Define null space of a matrix $A$. Let Then show that $v$ is in the null $A$. [5+0]

Determine the column of the matrix $A$ are linearly independent, where [5]

When two column vector in $\mathbb{R}^2$ are equal? Give an example. Compute $u + 3v$, $u - 2v$, where [5]

Find a unit vector $v$ of $u = (1, -2, 2, 3)$ in the direction of $u$. [5]

Prove that the two vectors $u$ and $v$ are perpendicular to each other if and only if the line through $u$ is perpendicular bisector of the line segment from $-u$ to $v$. [5]

Verify that $1k, (-2^k), 3k$ are linearly independent signals. [5]