2008

1) Determine $a$, $b$ and $c$ so that the matrix $A=\left(\begin{array}{ccc}0& 2b& c\\ a& b& -c\\ a& -b& c\end{array}\right)$ is orthogonal.

[10M]

2) If $\mathrm{S}$ and $\mathrm{T}$ are subspaces of ${\mathbb{R}}^4$ given by $S=\{(x,y,z,\omega )\in {\mathbb{R}}^4:2x+y+3z+\omega =0\}$ and $T=\{(x,y,z,\omega )\in {\mathbb{R}}^4:x+2y+z+3\omega =0\}$, find $\dim \mathrm{S}\cap \mathrm{T}$.

[10M]

3) Obtain the characteristic equation of the matrix $A=\left(\begin{array}{ll}1& 2\\ 2& -1\end{array}\right)$ and show that $A$ satisfies its characteristic equation. Hence determine the inverse of A.

[10M]

4) If $\mathrm{S}$ be a real skew-symmetric matrix of order $\mathrm{n}$, prove that the matrix $P={(I_n+S)}^{-1}(I_n-S)$ is orthogonal, where $I_n$ stands for the identity matrix of order $\mathrm{n}$.

[10M]

5) Find the row rank and column rank of the matrix
$A=\left(\begin{array}{llll}2& 1& 4& 3\\ 3& 2& 6& 9\\ 1& 1& 2& 6\end{array}\right)$
Hence determine the rank of $\mathrm{A}$.

[10M]

6) Reduce the equation $2y^2-2xy-2yz+2zx-x-2y+3z-2=0$ into canonical form and determine the nature of the quadric.

[10M]

2007

1) Suppose $U$ and $W$ are subspaces of the vector space ${\mathrm{R}}^4(\mathrm{R})$ generated by the sets

respectively. Determine dim $(U+W)$.

[10M]

2) Find the characteristic equation of the matrix

$A=\left[\begin{array}{rrr}2& -1& 1\\ -1& 2& -1\\ 1& -1& 2\end{array}\right]$ And verify that it is satisfied by $A$.

[10M]

3) Show that the solutions of the differential equation

is a subspace of the vector space of all real valued continuous functions.

[10M]

4) Show that vectors (0,2,-4),(1,-2,-1), 1,-4,3$)$ are linearly dependent. Also express $(0,2,-4)$ as a linear combination of $(1-2,-1)$ and 1,-4,3$)$.

[10M]

5) Is the matrix

similar over the field $\mathrm{R}$ to a diagonal matrix? Is $\mathrm{A}$ similar over the field $\mathrm{C}$ to a diagonal matrix?

[10M]

6) Determine the definiteness of the following quadratic form:

[10M]

2006

1) Let $W$ be the subspace of ${\mathbf{R}}^4$ generated by the vectors (1,-2,5,-3),(2,3,1,-4) and $(3,8,-3,-5)$. Find a basis and dimension of $W$.

[10M]

2) Find the symmetric matrix which corresponds to the following quadratic polynomial:

[10M]

3) Let

$\left(\begin{array}{ll}1& 4\\ 2& 3\end{array}\right)$.

Find all eigenvalues of $A$ and the corresponding eigenvectors.

[10M]

4) Let $\mathrm{H}=\left(\begin{array}{ccc}1& 1+i& 2\\ 1-i& 4& 2-3i\\ -2& 2+3i& 7\end{array}\right)$ be a Hermitian matrix. Find a non-singular matrix $\mathrm{P}$ such that $P^{\mathrm{\prime }}\mathrm{H}\overline{\mathrm{P}}$ is diagonal.

[10M]

5) Let $L:{\mathbf{R}}^3\to {\mathbf{R}}^2$ be a linear transfonnation for which we know that $L(1,0,0)=(2,-1)$, $L(0,1,0)=(3,1)$ and $L(0,0,1)=(-1,2)$. Find $L(-3,4,2)$.

[10M]

6) Let $\mathrm{L}:{\mathbf{R}}^3\to {\mathbf{R}}^2$ be defined by $L\left[\left[\begin{array}{l}x\\ y\\ z\end{array}\right]\right)=\left[\begin{array}{l}x+y\\ y-z\end{array}\right]$.

Let $\mathrm{S}=[{\mathrm{v}}_1,{\mathrm{v}}_2,{\mathrm{v}}_3]$ and $\mathrm{T}=[{\mathrm{w}}_1,{\mathrm{w}}_2]$ be bases for ${\mathrm{R}}^3$ and ${\mathrm{R}}^2$ respectively, where

Find the matrix of $L$ with respect to $S$ and $T$.

[10M]

2005

1) Let $V=P_2(R)$ be the vector space of polynomial functions over real of degree at most 3. Let $\mathrm{D}:\mathrm{V}\to \mathrm{V}$ be the differentiation operator defined by $\mathrm{D}(a_0+a_{1}x+a_2{x}^2+a_3{x}^3)=a_1+2a_{2}x+3a_3{x}^2,x\in \mathrm{R}$.
(i) Show that $\mathrm{B}\{1,\mathrm{x},{\mathrm{x}}^2,{\mathrm{x}}^3\}$ is a basis for $\mathrm{V}$.
(ii) Find the matrix $[D{]}_{\mathrm{B}}$ with respect to $\mathrm{B}$ of $\mathrm{D}$.
(iii) Show that ${\mathrm{B}}^{\mathrm{\prime }}\{1,(\mathrm{x}+1),(\mathrm{x}+1{)}^2,(\mathrm{x}+1{)}^3\}$ is a basis for $V$.
(iv) Find the matrix $[D{]}_{\mathrm{B}}$, with respect to ${\mathrm{B}}^{\prime }$ of D.
(v) Find the matrix $[\mathrm{D}{]}_{B^{\prime },B}$ of $\mathrm{D}$ relative to ${\mathrm{B}}^{\mathrm{\prime }}$ and $\mathrm{B}$.

[10M]

2) Find the eigen values and the corresponding eigenvectors of $A=\left(\begin{array}{ll}2& 1\\ 2& 3\end{array}\right)$.

[10M]

3) Show that the vectors $v_1=(1,1,1)$, $v_2=(0,1,1)$, $v_3=(0,0,1)$ form a basis for ${\mathbf{R}}^{(3)}$. Express $v=(3,1,4)$ as a linear combination of $v_1$, $v_2$ and $v_3$. Is the set $S=\{v,v_1,v_2,v_2,v_3\}$ linearly independent?

[10M]

4) Determine a non-singular matrix $P$ such that $P^{\mathrm{\prime }}AP$ is a diagonal matrix, where $P^{\mathrm{\prime }}$ denotes the transpose of $P,$ and $A=\left(\begin{array}{lll}0& 1& 2\\ 1& 0& 3\\ 2& 3& 0\end{array}\right)$.

[10M]

5) Determine a non-singular matirx $P$ such that $4P^{\mathrm{\prime }}AP$ is a diagonal matrix, where $P^{\mathrm{\prime }}$ denotes the transpose of $\mathrm{P}$, and $A=\left[\begin{array}{lll}0& 1& 2\\ 1& 0& 3\\ 2& 3& 0\end{array}\right]$.

[10M]

6) Show that the real quadratic form $\varphi =n(x_1^2+x_2^2+\dots .+x_n^2)-{(x_1+x_2+\dots +x_n)}^2$ in $n$ variables is positive semi-definite.

[10M]