2004

1) Let $\mathrm{U}$ and $\mathrm{W}$ be subspaces of ${\mathrm{R}}^3$ for which dim $\mathrm{U}=\mathrm{I},$ dim $\mathrm{W}=2$ and $\mathrm{U}\subset \mathrm{W}$. Show that ${\mathrm{R}}^3-\mathrm{U}+\mathrm{W}$ and $\mathrm{U}\cap \mathrm{W}=10\}$.

[10M]

2) Let $\{e_1,e_2,e_3\}$ be the standard basis of $R^3$ and $T$ be a linear transformation from $R^3$ into $R^2$ defined by $\mathrm{T}\left(e_1\right)=(2,3{)}^{\mathrm{\prime }},\mathrm{T}\left({\mathrm{e}}_2\right)=(1,2{)}^{\mathrm{\prime }}$ and $\mathrm{T}\left({\mathrm{e}}_3\right)=(-1,-4{)}^{\mathrm{\prime }}$ (where $\mathrm{\prime }$ means transpose).
(i) What is $\mathrm{T}(1,-2,-1)$?
(ii) What is the matrix of $T$ with respect to the standard bases of ${\mathrm{R}}^3$ and ${\mathrm{R}}^2$?

[10M]

3) Find a linear map $\mathrm{T}:{\mathrm{R}}^2\to {\mathrm{R}}^4$ whose range is generated by $(1,2,0,-4)$ and $(2,0,-1,-3)$ Also find basis and the dimension of the
(i) range $U$ of $T$. (ii) kernel $W$ of $T$.

[10M]

4) Find the eigen values and their corresponding eigen vectors of the matrix $\left(\begin{array}{lll}2& 4& 3\\ 2& 1& 2\\ 0& 0& 1\end{array}\right)$. Is the matrix diagonalizable?

[10M]

5) For what values of $a$ has the system of equations $$

[10M]

6) Determine an orthogonal matrix which reduces the quadratic form

to a canonical form. Also, identify the surface represented by $Q(x_1,x_2,x_3)=7$.

[10M]

2003

1) Let $V{P}_3(R)$ be the vector space of polynomial functions on reals of degree at most $3,$ Let $D$; $V\to V$ be the differention operator defined by $D(a_0+a_{1}x+a_2{x}^2+a_3{x}^3)=a_1+2a_{2}x+3a_3{x}^2$
(i) Show that $D$ is a linear transformation
(ii) Find kemel and image of $D$.
(iii) What are dimensions of $V_1$, ker D and image $D_4$?
(iv) Give relation among them of (iii).

[10M]

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

[10M]

3) Show that the vectors $(1,2,1)$, $(1,0,-1)$ and $(0,-3,2)$ form a basis for ${\mathrm{R}}^{(3)}$.

[10M]

4) Determine non-singular matrices $\mathrm{P}$ and $\mathrm{Q}$ such that the matrix $\mathrm{P}\mathrm{A}\mathrm{Q}$ is in canonical form, where

Hence find the rank of $\mathrm{A}$.

[10M]

5) Find the minimum polynomial of the matrix.
$A=\left(\begin{array}{lll}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right)$

and use it to determine whether $\mathrm{A}$ is similar to a diagonal matrix.

[10M]

6) Show that the quadratic form $2x^2-4xy+3xy+6y^2+6yz+8z^2$ in three variables is positive definite.

[10M]

2002

1) Let ${\mathrm{S}}_1,{\mathrm{S}}_2,\dots \dots .{\mathrm{S}}_{\mathrm{k}}$ be subspaces of a vector space $\mathrm{V}(\mathrm{F})$. Show that the following statements are equivalent:
(i) ${\mathrm{S}}_1+{\mathrm{S}}_2+\dots +{\mathrm{S}}_{\mathrm{k}}$ is a direct sum of $\mathrm{V}(\mathrm{F})$
(ii) $(S_1+S_2+\dots \dots .+S_i)\cap S_{,n1}=\{0\},$ for $i=1,2,\dots \dots \dots ,k-1$
(iii) $x_1+x_2+\dots \dots +x_k=0,x_1\in S_1,t=1,2$
$k\Rightarrow x_1=0$ for $t=1,2,\dots \dots \dots ,k$
(iv) $d(S,+S_2+\dots .+S_k)=d\left(S_1\right)+d\left(S_2\right)+\dots .+d\left(S_k\right)$.

[10M]

2) Given $A=\left[\begin{array}{l}{\displaystyle \frac{3}{2}}{\displaystyle \frac{1}{2}}\\ {\displaystyle \frac{1}{2}}{\displaystyle \frac{3}{2}}\end{array}\right]$ For which values of a does the vecter sequence ${\left\{y_n\right\}}_0^{-}$ defined by $y_n=(1+\alpha A+{\alpha }^2{A}^2)y_{n-1}n=1,2$.

[10M]

3) Show that if $\lambda$ is a non-zero eigen value of the non-singular $n$-square matrix $A$, then ${\displaystyle \frac{|A|}{\lambda }}$ is an eigen value of $adjA$.

Also given an example to prove that the eigen values of $\mathrm{A}\mathrm{B}$ are not necessarily the product of cigen values of $\mathrm{A}$ and that of $\mathrm{B}$.

[10M]

4) Given the liner transformation $Y=\left[\begin{array}{lll}1& 1& 0\\ 2& 3& 1\\ -2& 3& 5\end{array}\right]X$ show that it is singular and the images of the linearly independent vectors

$X_1=[1,1,1{]}^{\mathrm{\prime }}$
$X_i=[2,1,2{]}^{\mathrm{\prime }}$
$X_i=[1,2,3{]}^{\mathrm{\prime }}$
are linearly depchdent.

[10M]

5.(i) Calculate $f(A)=e^A-e^{-A}$ for

[5M]

TBC

5.(ii) The $n\times n$ matrix A satisfies $$ A^{4}=-1.6 A^{2}-0.641. $$ Show that $li{m}_{m\to \mathrm{\infty }}{A}^m$ exists and determine this limit.

[5M]

6) Reduce $A=\left[\begin{array}{llll}1& 3& 6& -1\\ 1& 4& 5& 1\\ 1& 5& 4& 3\end{array}\right]$ to normal form $\mathrm{N}$ and compute the matrices $\mathrm{P}$ and $Q$ such that $\mathrm{A}\mathrm{Q}=\mathrm{N}$

[10M]

2001

1) Let $\mathrm{V}$ be the vector space of polynomials in $\mathrm{x}$ with real coefficients of degree almost $2.$ Let $t_1,t_2,t_3$ be any 3 distinct real numbers. Define $L_1:V\to R$ by $L_1(f)=f(t),i=1,2,3$. Show that
(i) $L_4,L_2,L_3$ are linear functional on $\mathrm{V}$.
(ii) $\{L_1,L_2,L_3\}$ is a basis for the dual space $\mathrm{V}$ of $\mathrm{V}$.

[10M]

2) In the notation of (a) above, find a basis $B=\{P_1,P_2,P_3\}$ for $\mathrm{V}$ which is dual to $\{L_1,L_2,L_3\}$ and also express each $P\in V$ in terms of elements of $B$.

[10M]

3) Let $\mathrm{V}$ be the vector space of polynomials in $\mathrm{x}$ with complex coefficients. Define $T:V\to V$ by $(Tf)(x)=xf(x)$ and $U:V\to V$ by $U\left[\sum _{i=0}^n{c}_i{x}^{\mathrm{\prime }}\right]=\sum _{i=0}^{n-1}{c}_{i+1}{x}^{\mathrm{\prime }}$.
(i) Find ker $T$
(ii) Show that $U$ is linear
(iii) Show that $\mathrm{U}\mathrm{T}=\mathrm{I}$ and $\mathrm{T}\mathrm{V}\ne \mathrm{I},\mathrm{I}=$ Identity on $\mathrm{V}$.

[10M]

TBC

4) Let $A=\left(\begin{array}{lll}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right)$. Show that for every interger $n\ge 3A^n=A^{n+2}+A^2-I$ and hence find the matrix ${\mathrm{A}}^8$.

[10M]

5) Find the characteristic and minimal polynomials of $A=\left(\begin{array}{ccc}2& 0& 0\\ 1& 2& 0\\ 0& 0& -1\end{array}\right)$ and determine whether $A$ is diagonalisable.

[10M]

6) Let $A=\left(\begin{array}{ccc}1& 1& 2\\ -1& 2& 1\\ 0& 1& 3\end{array}\right),$ Find an invertible $3\times 3$ matrix $\mathrm{P}$ such that ${\mathrm{P}}^{\mathrm{\prime }}\mathrm{A}\mathrm{P}=\mathrm{D},\mathrm{D}$ is diagonal matrix. Find $D$ also.

[10M]

2000

1) Let $C(R)$ be the vector space of complex numbers over the field of real numbers. Under what conditions of the real numbers $\alpha ,\beta ,\gamma ,\delta$ in the set
$S=\{\alpha +i\beta ,\gamma +i\delta \},i=\sqrt{(-1)}$ do we have $L(s)=C(R)$, where $L(s)$ denotes the linear spans of $\mathrm{S}$? Justify your answer.

[10M]

2) Let $D:V(IR)\to V(IR):f(x)\to {\displaystyle \frac{df(x)}{dx}}$
$T:V(IR)\to V(IR):f(x)\to xf(x)$
be linear transformations on $\mathrm{V}$ (IR), the vector space of all polynomials in an indeterminate $\mathrm{X}$ with real coefficients.
Show that
(i) $\mathrm{D}\mathrm{T}-\mathrm{T}\mathrm{D}=\mathrm{I}$ the identity operator
(ii) $(TD{)}^2=T{D}^2+TD$

[10M]

3) Find $A^{-2}$ when $A=\left[\begin{array}{ccc}1& 2& 0\\ 2& -1& 0\\ 0& 0& -1\end{array}\right]$ by using Cayley-Hamilton theorem.

[10M]

4) Show that every square matrix can be expressed uniquely as $A+iB$ where $A$, $B$ are Hermitian.

[10M]

56) Reduce the quadratic form $x^2+4y^2+9z^2+t^2-12yz+6zx4xy-2xt-6zt$ to canonical form and find its rank and signature.

[10M]