IAS PYQs 1
2000
1) Let \(V\) be a vector space over \(\mathrm{R}\) and Let \(\mathrm{T}=\{(\mathrm{x}, \mathrm{y}) \mid \mathrm{x}, \mathrm{y}, \in \mathrm{V}\}\)Define addition in T componentwisc and scalar multiplication by a complex number \(\alpha+\mathrm{i} \beta\) by \((\alpha+i \beta)(x, y)=(\alpha x-\beta y, \beta x+\alpha y)\) \(\forall \alpha, \beta \in R\). Show that \(\mathrm{T}\) is a vector space over \(C\).
[10M]
2) Show that if \(\lambda\) is a characteristic root of a non-singular matrix \(A,\) then \(\lambda^{-1}\) is a characteristic root of \(A^{-1}\).
[10M]
3) Use the Mean value theorem to prove \(\dfrac{2}{7}<\log 1.4<\dfrac{2}{5}\).
[10M]
4) Show that \(\iint x^{2 l-1} y^{2 m-1} d x d y=\dfrac{1}{4} r^{2(l+m)} \dfrac{ \Gamma{l} \Gamma{m}}{\Gamma(l+m+1)}\) for all positive values of \(x\) and \(y\) lying inside the circle \(x^{2}+y^{2}=r^{2}\).
[10M]
5) Find the equations of the planes bisecting the angles between the planes \(2 x-y-2 z-3=0\) and \(3 x+4 y+1=0\) and specify the one which bisects the açute angle.
[10M]
6) Find the equation to the common conjugate diameters of the conics \(x^{2}+4 x y+6 y^{2}=1\) and \(2 x^{2}+6 x y+9 y^{2}=1\).
[10M]
7) Prove that a real symmetnc matrix A is positive definite if and only \(\mathrm{A}=\mathrm{BB}^{t}\) for some non-singular matrix \(\mathrm{B}\). Show also that \(A=\left(\begin{array}{lll} 1 & 2 & 3 \\ 2 & 5 & 7 \\ 3 & 7 & 11 \end{array}\right)\) is positive definite and find the matrix \(B\) such that \(A=B B^{t}\). Here \(B^{t}\) stands for the transpose of \(\mathrm{B}\).
[10M]
8) Prove that a system \(\mathrm{AX}=\mathrm{B}\) of \(\mathrm{n}\) non-homogeneous equations in \(\mathrm{n}\) unknowns has a unique solution provided the coefficient matrix is non singular.
[10M]
9) Prove that two similar matrices have the same characteristic roots. Is its converse true? Justify your claim.
[10M]
2.d) Reduce the equation \(x^{2}+y^{2}+z^{2}-2 x y-2 y z+2 z x+x-y-2 z+6=0\) into canonical form and determine the nature of the quadratic.
[10M]
1999
1) Let \(V\) be the vector space of functions from \(R\) to \(R\) (the real numbers).Show that \(f,g,h\) in \(V\) are linearly independent were \(f(t)=e^{2t}\), \(g(t)=t^2\;and\;h(t)=t\).
[10M]
2) If the matrix of a linear transformation \(T\) on \(V_2(R)\) with repect to the basis \(B=\{(1,0),(0,1)\}\) is \(\begin{bmatrix} 1& 1\\ 1&1 \end{bmatrix}\), then what is the matrix of \(T\) with respect to the ordered basis \(B_1=\{(1,1),(1,-1)\}\)?
[10M]
3) Diagonalize the matrix \(A=\begin{bmatrix} 4& 2&2\\ 2& 4&2\\ 2&2&4 \end{bmatrix}\)
[10M]
4) Test for congruency of the matrices \(A=\begin{bmatrix} 1& 0\\ 0& -1 \end{bmatrix}\) and \(B=\begin{bmatrix} 0& i\\ -i& 0 \end{bmatrix}\).Prove that \(A^{2m}=B^{2m}=I\) where \(m\) and \(n\) are positive integers.
[10M]
5) If \(A\) is a skew symmetric matrix of order \(n\), prove that \((I-A)(I+A)^{-1}\) is orthogonal.
[10M]
6) Test for the positive definiteness of the quadratic form \(2x^2+y^2+2z^2+2xy-2zx\)
[10M]
1998
1) Given two linearly independent vectors (1,0,1,0) and (0,-1,1,0) of \(R^4\), find a basis \(R^4\) which includes these two vectors.
[10M]
2) If \(V\) is a finite dimensional vector space over \(R\) and if f and g are two linear transformation from \(V\) to \(R\) such that \(f(v)=0\) implies \(g(v)=0\) then prove that \(g=\lambda\)f for soome \(\lambda\) in \(R\).
[10M]
3) Let T:\(R^3\to R^3\) be defined by \(T(x_1,x_2,x_3)=(x_2,x_3-cx_1-bx_2-ax_3)\), where a,b,c are fixed real numbers.Show that T is a linear transformation of \(R^3\) and that \(A^3+aA^2+bA+cI=0\) ,where \(A\) is the matrix of \(T\) with respect to standard basis of \(R^3\)
[10M]
4) If \(A\) and \(B\) are two matrices of order \(2\times2\) such that \(A\) is skew Hermitian and \(AB=B\), then show that \(B=0\).
[10M]
5) If \(T\) is a complex matrix of order \(2\times2\) such that T=tr \(T^2=0\), then show that \(T^2=0\).
[10M]
6) Prove that a necessary and sufficient condition for a \(n\times n\) real matrix \(A\) to be similar to a diagonal matrix is that the set of characteristic vectors of \(A\) includes a set of \(n\) linearly independent vectors.
[10M]
7) Let \(A\) be an \(m\times n\) matrix. Then show that the sum of the real and nullity of \(A\) is \(n\).
[10M]
8) Find all real \(2\times2\) matrices \(A\) whose characteristic roots are real and which satisfy \(AA’=I\).
[10M]
9) Reduce to diagonal martrix by rational congruent transformation the symmetric matrix \(A= \begin{bmatrix} 1& 2& -1\\ 2& 0& 3\\ -1& 3& 1 \end{bmatrix}\)
[10M]
1997
1) Let \(\mathrm{V}\) be the vector space of polynomials over \(\mathrm{R}\). Find a basis and dimension of the subspace \(\mathrm{W}\) of \(\mathrm{V}\) spanned by the polynomials
\(\mathrm{v}_{1}=\mathrm{t}^{3}-2 \mathrm{t}^{2}+4 \mathrm{t}+1\), \(\mathrm{v}_{2}=2\mathrm{t}^{3}-3 \mathrm{t}^{2}+9 \mathrm{t}-1\), \(\mathrm{v}_{3}=\mathrm{t}^{3}\mathrm{t}-5\), \(\mathrm{v}_{4}=2 {t}^{3}-5 \mathrm{t}^{2}+7 \mathrm{t}+5\).
[10M]
2) Verify that the transformation defined by \(\mathrm{T}\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right)=\left(\mathrm{x}_{1}+\mathrm{x}_{2}, \mathrm{x}_{1}-\mathrm{x}_{2}, \mathrm{x}_{2}\right)\) is a linear transformation from \(\mathrm{R}^{2}\) into \(\mathrm{R}^{3}\). Find its range, null space and nullity.
[10M]
3) Let \(\mathrm{V}\) be the vector space of \(2 \times 2\) matrices over \(\mathrm{R}\). Determine whether the matrices \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) \(\in \mathrm{V}\) are dependent where \(A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 1 \end{array}\right], B=\left[\begin{array}{cc} 3 & -1 \\ 2 & 2 \end{array}\right], C=\left[\begin{array}{cc} 1 & -5 \\ -4 & 0 \end{array}\right]\)
[10M]
4) Let a square matrix A of order \(n\) be such that each of its diagonal elements is \(\mu\) and each of its off diagonal elements is 1 . If \(B=\lambda A\) is orthogonal, determine the values of \(\lambda\) and \(\mu\).
[10M]
5) Show that \(A=\left[\begin{array}{ccc}2 & -1 & 0 \\ -1 & 2 & 0 \\ 2 & 2 & 3\end{array}\right]\) is diagoalisable over \(\mathrm{R}\) and find a matrix \(\mathrm{P}\) such that \(\mathrm{P}^{-1} \mathrm{AP}\) is diagonal. Hence determine \(A^{25}\).
[10M]
6) Let \(A=\left[a_{j}\right]\) be a square matrix of order \(n\) such that \(\left[a_{i j}\right] \leq M \forall i, j=1,2, \ldots, n\). Let \(\lambda\) be an eigen-value of \(A\), Show that \(\vert \lambda \vert \leq \mathrm{nM}\).
[10M]
7) Define a positive definite matrix. Show that a positive definite matrix is always non-singular. Prove that its converse does not hold.
[10M]
8) Find the characteristic roots and their corresponding vectors for the matrix \(\left[\begin{array}{ccc} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{array}\right]\)
[10M]
9) Find an invertible matrix \(\mathrm{P}\) which reduces \(Q(x, y, z)=2 x y+2 y z+2 x\) to its canonical form.
[10M]
1996
1) \(\mathrm{R}^{4},\) let \(\mathrm{W}_{1}\) be the space generated by (1,1,0,-1),(2,6,0) and (-2,-3,-3,1) and let \(\mathrm{W}_{2}\) be the space generated by \((-1,-2,-2,2),(4 ;,6,4,-6)\) and \((1,3,4,-3)\) . Find a basis for the space \(W_{1}+W_{2}\)
[12M]
2) Let \(\mathrm{V}\) be a finite dimensional vector space and \(\mathrm{v} \in \mathrm{V}, \mathrm{v} \neq 0 .\) Show that there exists a linear functional \(f\) on \(\mathrm{V}\) such that \(f(v) \neq 0 .\)
[12M]
3) Let \(\mathrm{V}=\mathrm{R}^{3}\) and \(\mathrm{v}_{1}, \mathrm{v}_{2}, \mathrm{v}_{3}\) be a basis of \(\mathrm{R}^{3}\). Let \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{V}\) be a linear transformation such that \(\mathrm{T}\left(\mathrm{v}_{1}\right)=\mathrm{v}_{1}+\mathrm{v}_{2}+\mathrm{v}_{3}, 1 \leq \mathrm{i} \leq 3 .\) By writing the matrix of \(\mathrm{T}\) with respect to another basis, show that the matrix \(\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{array}\right]\) is similar to \(\left[\begin{array}{lll}3 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]\)
[12M]
4) Let \(\mathrm{V}=\mathrm{R}^{3}\) and \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{V}\) be the linear map defined by \(\mathrm{T}(\mathrm{x}, \mathrm{y}, z)=(\mathrm{x}+2,-2 \mathrm{x}+\mathrm{y},-\mathrm{x}+2 \mathrm{y}+\mathrm{z})\) What is the matrix of T with respect to the basis (1,0,1),(-1,1,1) and (0,1,1)\(?\) Using this matrix, write down the matrix of T with respect to the basis (0,1,2),(-1,1,1) and (0,1,1).
[15M]
5) Let \(\mathrm{V}\) and \(\mathrm{W}\) be finite dimensional vector spaces such that \(\mathrm{dim} \mathrm{V} \geq \mathrm{dim} \mathrm{W}\). Show that there is always a linear map of \(\mathrm{V}\) onto \(\mathrm{W}\).
[15M]
6) Solve
\[\begin{array}{l} x+y-2 z=1 \\ 2 x-7 z=3 \\ x+y-y=5 \end{array}\]by using Cramer’s rule.
[15M]
7) Find the inverse of the matrix \(\left[\begin{array}{llll} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{array}\right]\) by computing its characteristic polynomial.
[15M]
8) Let \(A\) and \(B\) be \(n \times n\) matrices such that \(A B=B A\). Show that \(A\) and \(B\) have a common characteristic vector.
[15M]
9) Reduce to canonical form the orthogonal matrix
\[\left[\begin{array}{ccc} 2 / 3 & -2 / 3 & 1 / 3 \\ 2 / 3 & 1 / 3& -2 / 3 \\ 1 / 3 & 2 / 3 & 2 / 3 \end{array}\right]\][15M]
1995
1) Let \(T\) be the linear operator in \(\mathrm{R}^{3}\) defined by \(\mathrm{T}\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}\right)=\left(3 \mathrm{x}_{1}+\mathrm{x}_{3},-2 \mathrm{x}_{1}+\mathrm{x}_{2,}-\mathrm{x}_{1}+2 \mathrm{x}_{2}+4 \mathrm{x}_{3}\right)\) What is the matrix of \(T\) in the standard ordered basis for \(R^{3} ?\) What is a basis of range space of T and a basis of null space of T?
[20M]
2) Let \(A\) be a square matrix of order \(\mathrm{n}\). Prove that \(\mathrm{AX}=\mathrm{b}\) bas a solution if and only if be \(\mathrm{R}^{\prime \prime}\) is orthogonal to all solutions Y of the svstem \(A^{\top} Y=0\).
[20M]
3) Define a similar matrix. Prove that the characteristic equation of two similar matrices is the same. Let 1,2,3 be the eigen-values of a matrix. Write down such a matrix. Is such a matrix unique?
[20M]
4) Show that \(A=\left[\begin{array}{ccc}5 & -6 & -6 \\ -1 & 4 & 2 \\ 3 & -6 & -4\end{array}\right]\) is diagonalizable and hence determine \(A^{3}\).
[20M]
5) Let \(A\) and \(B\) be matrices of order \(n\) Prove that if \((I-A B)\) is invertible, then \((I-B A)\) is also invertible and \((1-B A)^{-1}=1+B(1-A B)^{-1} A\)
[20M]
6) If \(a\) and \(b\) are complex numbers such that \(\vert \mathrm{b} \vert =\mathrm{I}\) and \(\mathrm{H}\) is a Hermitian matrix, show that the eigen-values of $$aI+$bH$ lie on a straight line in the complex plane.
[20M]
7) Let \(A\) be a symmefric matrix. Show that \(A\) is positive definite if and only if its eigen-values are all positive
[20M]
8) Let \(A\) and \(B\) be square matrices of order \(n\). Show that \(A B\) -BA can never be equal to unit matrix.
[20M]
9) Let \(\mathrm{A}=\left[a_{i j}\right] ; \mathrm{i}, \mathrm{j}=1,2, \ldots, \mathrm{n}\) and \(\left|a_{n}\right| > \sum_{j\neq i}\left|a_{i j}\right| j=1,2, \ldots, n\) for every \(i=1,2...,n\). Show that A is non-singualr matrix. Hence or otherwise prove that the eigen-values of A lie in the discs \(\left|\lambda-a_{n}\right| \leq \sum_{j\neq i}\left|a_{i j}\right| j=1,2, \ldots, n\) in the complex plane.
[20M]