1994

1) Show that $f_1(t)=1,f_2(t)=t-2,f_3(t)=(t-2{)}^2$ form a basis of $P_2$, the space of polynomials with degree $\le 2$. Express $3t^2-5t+4$ as a linear combination of $f_1,f_2,f_3$.

[10M]

2) If $T:V_4(R)\to V_3(R)$ is a linear transformation defined by $T(a,b,c,d)=(a-b+c+d,a+2c-d,a+b+3c-3d)$

For $a$, $b$, $d\in R$, then verify that Rank $T+$ Nullity $T=\dim V_4(R)$.

[10M]

3) If $T$ is an operator on $R_3$ whose basis is $B=\{(1,0,0),(0,1,0),(0,0,1)\}$ such that
$[T:B]=\left[\begin{array}{ccc}0& 1& 1\\ 1& 0& -1\\ -1& -1& 0\end{array}\right]$ find the matrix of $T$ with respect to a basis $B_1=\{(0,1,-1),(1,-1,1),(-1,1,0)\}$.

[10M]

4) If $A=\left[a_{ij}\right]$ is an $n\times n$ matrix such that $a_{ii}=n,a_{ij}=r$ if $i\ne j,$ show that $[A-(n-r)H][A-(n-r+nr)I]=0$
Where $b_{ii}=1,b_{ij}=\rho$ when $i\ne j$ and $\rho \ne 1,\rho \ne {\displaystyle \frac{1}{1-n}}$.

[10M]

5) Prove that the eigen vectors corresponding to distinct eigen values of a square matrix are linearly independent.

[10M]

6) Determine the eigen values and eigen vectors of the matrix $A=\left[\begin{array}{lll}3& 1& 4\\ 0& 2& 6\\ 0& 0& 5\end{array}\right]$

[10M]

7) Show that a matrix congruent to a skew-symmetric matrix is skew-symmetric. Use the result to prove that the determinant of skew-symmetric matrix of even order is the square of a rational function of its elements.

[10M]

8) Find the rank of the matrix $\left[\begin{array}{cccc}0& c& -b& a^{\mathrm{\prime }}\\ -c& 0& a& b^{\mathrm{\prime }}\\ b& -a& 0& c^{\mathrm{\prime }}\\ -a^{\mathrm{\prime }}& -b^{\mathrm{\prime }}& -c^{\mathrm{\prime }}& 0\end{array}\right]$ where $a{a}^{\mathrm{\prime }}+b{b}^{\mathrm{\prime }}+c{c}^{\mathrm{\prime }}=0$ and $a,b,c$ are all positive integers.

[10M]

9) Reduce the following symmetric matrix to a diagonal form and interpret the result in terms of quadratic forms: $A=\left[\begin{array}{ccc}3& 2& -1\\ 2& 2& 3\\ -1& 3& 1\end{array}\right]$

[10M]

1993

1) Show that the set $S=\{(1,0,0),(1,1,0),(1,1,1),(0,1,0)\}$ spans the vector space $R^3(R)$ but it is not a basis. set.

[10M]

2) Define rank and nullity of a linear transformation $T$. If $V$ be a finite dimensional vector space and $T$ a linear operator on $v$ such that rank $T^2=$ rank $T$, then prove that the null space of $T=$ the null space of $T^2$ and the intersection of the range space and null space to $T$ is the zero subspace of $V$.

[10M]

3) If the matrix of a linear operator $T$ on $R^2$ relative to the standard basis {(1,0),(0,1)} is $\left(\begin{array}{ll}1& 1\\ 1& 1\end{array}\right)$, what is the matrix of $T$ relative to the basis $B=\{(1,1),(1,-1)\}$?

[10M]

4) Prove that the inverse of $\left(\begin{array}{ll}A& O\\ B& C\end{array}\right)$ is

where $A$, $C$ are non-singular matrices and hence find the inverse of

[10M]

5) If $A$ be an orthogonal matrix with the property that- 1 is not an eigen value, then show that a is expressible as ( $f-S)(S+S{)}^{-1}$ for some suitable skew-symmetric matrix $S$.

[10M]

TBC

6) Show that any iwo eigen vectors corresponding to two distinct eigen values of
(i) Hermitian matrix
(ii) Unitary matrix are orthogonal.

[10M]

7) A matrix $B$ or order $n\times n$ is of the form $\lambda$ A where $\lambda$ is a scalar and $A$ has unit elements every where except in the diagonal which has elements $\mu$. Find $\lambda$ and $\mu$ so that $B$ may be orthogonal.

[10M]

8) Find the rank of the matrix $A=\left(\begin{array}{cccc}1& -1& 3& 6\\ 1& 3& -3& -4\\ 5& 3& 3& 11\end{array}\right)$ by reducing it to canonical form.

[10M]

9) Determine the following form as definite, semi-definite or indefinite:

[10M]

1992

1) Let $\mathrm{V}$ and $\mathrm{U}$ be vector spaces over the field $\mathrm{K}$ and let $\mathrm{V}$ be of finite dimension Let $\mathrm{T}:\mathrm{V}\to$ $\mathrm{U}$ be a linear Map. Prove that $\dim \mathrm{V}=\dim \mathrm{R}(\mathrm{T})+\dim \mathrm{N}(\mathrm{T})$.

[10M]

2) Let $\mathrm{S}=\{(\mathrm{x},\mathrm{y},z)/\mathrm{x}+\mathrm{y}+z=0\},\mathrm{x},\mathrm{y},z$ being real. Prove that $\mathrm{S}$ is a subspace of $R^3$. Find a basis of $S$.

[10M]

3) Verify which of the following are linear transformations:
(i) $T:B\to B^2$ defined by $T(x)=(2x,-x)$
(ii) $T:{\underset{\_}{R}}^2\to R^3$ defined by $T(x,y)=(xy,y,x)$
(iii) $T:B^2\to R^3$ defined by $T(x,y)=(x+y,y,x)$
(iv) $T:R\to B^2$ defined by $T(x)=(1,-1)$

[10M]

TBC

4) Let $\mathrm{T}:{\mathrm{M}}_{21}\to {\mathrm{M}}_{23}$ be a linear transformation defined by (with usual notations) $T\left(\begin{array}{l}1\\ 0\end{array}\right)=\left(\begin{array}{lll}2& 1& 3\\ 4& 1& 5\end{array}\right),T\left(\begin{array}{l}1\\ 1\end{array}\right)=\left(\begin{array}{lll}6& 1& 0\\ 0& 0& 2\end{array}\right)$.

Find $T\left({\displaystyle \frac{x}{y}}\right)$

[10M]

5) For what values of $n$ do the following equations
$\begin{array}{l}x+y+z\le 1\\ x+2y+4z=n\\ x+4y+10z=n^2\end{array}$

have a solution? Solve them completely in each case.

[10M]

6) Prove that a necessary and sufficient condition of a real quadratic form ${\mathrm{X}}^{\mathrm{\prime }}\mathrm{A}\mathrm{X}$ to be positive definite is that the leading principal minors of $A$ are all positive.

[10M]

7) State Cayley-Hamilton theorem and use it to calculate the inverse of the matrix $\to \left[\begin{array}{ll}2& 1\\ 4& 3\end{array}\right]$

[10M]

8) Transform the following to the diagonal forms and give the transformation employed:

[10M]

9) Prove that the characteristic roots of a Hermitian matrix are all real and a characteristic root of a skew-Hermitian matrix is either zero or a pure imaginary number.

[10M]

1991

1) Let $\mathrm{V}(\mathrm{R})$ be the real vector space of all $2\times 3$ matrices with real entries. Find a basis for $\mathrm{V}(R)$. What is the dimension of $\mathrm{V}(R)$?

2) Let C be the field of complex numbers and let T be the function from ${\mathbb{C}}^3$ into ${\mathrm{C}}^3$ defined by $\mathrm{T}({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3)=({\mathrm{x}}_1-{\mathrm{x}}_2+2{\mathrm{x}}_3,2{\mathrm{x}}_1+{\mathrm{x}}_{2^7}-{\mathrm{x}}_1-2{\mathrm{x}}_2+2{\mathrm{x}}_3)$.
(i) Verify that $\mathrm{T}$ is a linear transformation.
(ii) If $(a,b,c)$ is a vector in $C^3,$ what are the conditions on $\mathrm{a},\mathrm{b}$ and $\mathrm{c}$ so that the vector be in the range of $T$? what is the rank of $T$? (iii) What are the conditions on $a$, $b$ and $c$ so that $(a,b,c)$ is in the null space of $T?$ what is the nullity of $T$?

3) If $A=\left(\begin{array}{rr}1& 2\\ -1& 3\end{array}\right)$, express $A^6-4A^5+8A^4-12A^3+14{\mathrm{A}}^2$ as a linear polynomial in $\mathrm{A}$.

4) Let $T$ be the linear transformation from $R^2$ to itself defined by $\mathrm{T}({\mathrm{x}}_1,{\mathrm{x}}_2)=(-{\mathrm{x}}_2,{\mathrm{x}}_1)$.
(i) What is the matrix of $\mathrm{T}$ in the standard ordered basis for ${\mathrm{R}}^2$?
(ii) What is the matrix of $\mathrm{T}$ in the ordered basis $\beta =\{{\alpha }_1,{\alpha }_2\}$ where ${\alpha }_1=(1,2)$ and ${\alpha }_2=(1,-1)$?

5) Determine a non-singular matrix $\mathrm{P}$ such that ${\mathrm{P}}^{\mathrm{t}}\mathrm{A}\mathrm{P}$ is a diagonal matrix, where $\mathrm{A}=\left(\begin{array}{lll}0& 1& 2\\ 1& 0& 3\\ 2& 3& 0\end{array}\right)$.

Is the matrix $A$ congruent to a diagonal matrix? Justify.

6) Reduce the matrix $\left[\begin{array}{cccc}1& 0& 0& 0\\ -1& 1& 0& 0\\ 1& 1& 1& 0\\ 1& 1& 1& 1\end{array}\right]\left[\begin{array}{cccc}1& 3& 4& -5\\ -2& -5& -10& 16\\ 5& 9& 33& -68\\ 4& 7& 30& -78\end{array}\right]$

to echelon form by elementary row operations.

7) $\mathrm{U}$ is an n-rowed unitary matrix such that $|I-U|\ne 0$. Show that the matrix $\mathrm{H}$ defined by setting iH $=(\mathrm{I}+\mathrm{U})$ $(\mathrm{I}-\mathrm{U}{)}^{-1}$ is hermitian. Also show that if ${\mathrm{e}}^{{\mathrm{i}\mathrm{t}}_1},{\mathrm{e}}^{{\mathrm{i}\mathrm{t}}_2},\dots \dots \dots {\mathrm{e}}^{{\mathrm{i}\mathrm{t}}_n}$ be the eigen values of $\mathrm{U},$ the eigen values of $\mathrm{H}$ are $\cot ({\alpha }_1/2),\cot ({\alpha }_2/2),\dots \dots \dots$ $\cot ({\alpha }_n/2)$.

8) Let A be an n-rowed square matrix with distinct eigenvalues. Show that if $\mathrm{A}$ is non-singular, then there exists $2^n$ matrices $X$ such that $X^2=A$. What happens in case $\mathrm{A}$ is a singular matrix?

1989

1) Find a basis for the null space of the matrix. $A=\left[\begin{array}{ccc}3& 1& -1\\ 0& 1& 2\end{array}\right]$.

2) If $W$ is a subspace of a finite dimensional space $V$, then prove that $\dim \mathrm{V}/\mathrm{W}=\mathrm{dimV}-\mathrm{dimW}$

3) Show that all vectors $(x_1,x_2,x_3,x_4)$ in the vector space ${\mathrm{V}}_4(\mathrm{R})$ which obey ${\mathrm{x}}_4-{\mathrm{x}}_3={\mathrm{x}}_2-{\mathrm{x}}_1$ form a subspace $\mathrm{V}$. Show that further that $\mathrm{V}$ is spanned by $(1,0,0,-1)$, $(0,1,0,1)$, $(0,0,1,1)$.

4) Let $\mathrm{P}$ be a real skew symmetric matrix and $\mathrm{I},$ the corresponding unit matrix. Show that the matrix $I-P$ is non-singular. Also show that the matrix $Q=$ $(\mathrm{I}+\mathrm{P})(\mathrm{I}-\mathrm{P}{)}^{-1}$ is orthogonal.

5) Show that an $n\times n$ matrix $A$ is similar to a diagonal matrix if and only if the set of characteristic vectors of $A$ includes a set of $n$ linearly independent vectors.

6) Let ${\mathrm{r}}_1$ and ${\mathrm{r}}_2$ be distinct characteristic roots of a matrix $\mathrm{A},$ and let ${\xi }_i$ be a characteristic vector of $\mathrm{A}$ corresponding to ${\mathrm{r}}_i(\mathrm{i}=1,2).$ If $\mathrm{A}$ is Hermitian, then show that ${\xi }_1{\xi }_2=0$

7) Show that $\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$=$\left[\begin{array}{cc}1& -\tan \theta /2\\ \tan \theta /2& 1\end{array}\right]{\left[\begin{array}{cc}1& \tan \theta /2\\ -\tan \theta /2& 1\end{array}\right]}^{-1}$

8) Verify cayley - Hamilton theorem for the matrix $A=\left[\begin{array}{ll}0& 1\\ -2& 3\end{array}\right]$