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 \(\left.\mathrm{U} \cap \mathrm{W}=10\right\}\).

[10M]

2) Let \(\left\{e_{1}, e_{2}, e_{3}\right\}\) be the standard basis of \(R^{3}\) and \(T\) be a linear transformation from \(R^{3}\) into \(R^{2}\) defined by \(\mathrm{T}\left(e_{1}\right)=(2,3)^{\prime}, \mathrm{T}\left(\mathrm{e}_{2}\right)=(1,2)^{\prime}\) and \(\mathrm{T}\left(\mathrm{e}_{3}\right)=(-1,-4)^{\prime}\) (where \({\prime}\) 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 \(\mathrm{T}: \mathrm{R}^{2} \rightarrow \mathrm{R}^{4}\) 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 \(\) \begin{array}{l} x+2 y+z=1
\alpha x+4 y+2 z=2
4 x-2 y+2 \alpha z=-1 \end{array} \(\) (i) a unique solution?
(ii) infintiely many solutions?
(iii) no solution?

[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 \(D\); \(V \rightarrow V\) be the differention operator defined by \(D\left(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}\right)=a_{1}+2 a_{2} x+3 a_{3} x^{2}\)
(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 \(\mathrm{PAQ}\) 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 \(2 x^2-4 x y+3 x y+6 y^{2}+6 y z+8 z^{2}\) in three variables is positive definite.

[10M]

2002

1) Let \(\mathrm{S}_{1}, \mathrm{S}_{2}, \ldots \ldots . \mathrm{S}_{\mathrm{k}}\) be subspaces of a vector space \(\mathrm{V}(\mathrm{F})\). Show that the following statements are equivalent:
(i) \(\mathrm{S}_{1}+\mathrm{S}_{2}+\ldots+\mathrm{S}_{\mathrm{k}}\) is a direct sum of \(\mathrm{V}(\mathrm{F})\)
(ii) \(\left(S_{1}+S_{2}+\ldots \ldots .+S_{i}\right) \cap S_{, n 1}=\{0\},\) for \(i=1,2, \ldots \ldots \ldots, k-1\)
(iii) \(x_{1}+x_{2}+\ldots \ldots+x_{k}=0, x_{1} \in S_{1}, t=1,2\)
\(k \Rightarrow x_{1}=0\) for \(t=1,2, \ldots \ldots \ldots, k\)
(iv) \(d\left(S,+S_{2}+\ldots .+S_{k}\right)=d\left(S_{1}\right)+d\left(S_{2}\right)+\ldots .+d\left(S_{k}\right)\).

[10M]

2) Given \(A=\left[\begin{array}{l}\dfrac{3}{2} \dfrac{1}{2} \\ \dfrac{1}{2} \dfrac{3}{2}\end{array}\right]\) For which values of a does the vecter sequence \(\left\{y_{n}\right\}_{0}^{-}\) defined by \(y_{n}=\left(1+\alpha A+\alpha^{2} A^{2}\right) y_{n-1} n=1,2\).

[10M]

3) Show that if \(\lambda\) is a non-zero eigen value of the non-singular \(n\)-square matrix \(A\), then \(\dfrac{\vert A \vert}{\lambda}\) is an eigen value of \(adj A\).

Also given an example to prove that the eigen values of \(\mathrm{AB}\) 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

\(X_{1}=[1, 1,1]^{\prime}\)
\(X_{i}=[2,1,2]^{\prime}\)
\(X_{i}=[1,2,3]^{\prime}\)
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 \(lim_{m \to \infty} A^m\) exists and determine this limit.

[5M]

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

[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 \(L_{1}: V \rightarrow R\) 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) \(\left\{L_{1}, L_{2}, L_{3}\right\}\) is a basis for the dual space \(\mathrm{V}\) of \(\mathrm{V}\).

[10M]

2) In the notation of (a) above, find a basis \(B=\left\{P_{1}, P_{2}, P_{3}\right\}\) for \(\mathrm{V}\) which is dual to \(\left\{L_{1}, L_{2}, L_{3}\right\}\) 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 \(T: V \rightarrow V\) by \((T f)(x)=x f(x)\) and \(U: V \rightarrow V\) by \(U\left[\sum_{i=0}^{n} c_{i} x^{\prime}\right]=\sum_{i=0}^{n-1} c_{i+1} x^{\prime}\).
(i) Find ker \(T\)
(ii) Show that \(U\) is linear
(iii) Show that \(\mathrm{UT}=\mathrm{I}\) and \(\mathrm{TV} \neq \mathrm{I}, \mathrm{I}=\) 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 \(n \geq 3 A^{n}=A^{n+2}+A^{2}-I\) 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 \(\mathrm{P}^{\prime} \mathrm{AP}=\mathrm{D}, \mathrm{D}\) 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 \(D: V(I R) \rightarrow V(I R): f(x) \rightarrow \dfrac{d f(x)}{d x}\)
\(T: V(I R) \rightarrow V(I R): f(x) \rightarrow x f(x)\)
be linear transformations on \(\mathrm{V}\) (IR), the vector space of all polynomials in an indeterminate \(\mathrm{X}\) with real coefficients.
Show that
(i) \(\mathrm{DT}-\mathrm{TD}=\mathrm{I}\) the identity operator
(ii) \((T D)^{2}=T D^{2}+T D\)

[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 \(A+i B\) where \(A\), \(B\) are Hermitian.

[10M]

56) Reduce the quadratic form \(x^{2}+4 y^{2}+9 z^{2}+t^{2}-12 y z+6 z x \quad 4 x y-2 x t-6 z t\) to canonical form and find its rank and signature.

[10M]