IFoS PYQs 4
2008
1) Determine \(a\), \(b\) and \(c\) so that the matrix \(A=\left(\begin{array}{ccc}0 & 2 b & c \\ a & b & -c \\ a & -b & c\end{array}\right)\) is orthogonal.
[10M]
2) If \(\mathrm{S}\) and \(\mathrm{T}\) are subspaces of \(\mathbb{R}^{4}\) given by \(S=\left\{(x, y, z, \omega) \in \mathbb{R}^{4}: 2 x+y+3 z+\omega=0\right\}\) and \(T=\left\{(x, y, z, \omega) \in \mathbb{R}^{4}: x+2 y+z+3 \omega=0\right\}\), find \(\operatorname{dim} \mathrm{S} \cap \mathrm{T}\).
[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 \(P=\left(I_{n}+S\right)^{-1}\left(I_{n}-S\right)\) 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 \(2 y^{2}-2 x y-2 y z+2 z x-x-2 y+3 z-2=0\) 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
\[\begin{array}{l} B_{1}=\{(1,1,0,-1),(1,2,3,0),(2,3,3,-1)\} \\ B_{2}=\{(1,2,2,-2),(2,3,2,-3),(1,3,4,-3)\} \end{array}\]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
\[2 \dfrac{d^{2} y}{d x^{2}}-9 \dfrac{d y}{d x}+2 y=0\]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
\[A=\left[\begin{array}{rrr} 6 & -3 & -2 \\ 4 & -1 & -2 \\ 10 & -5 & -3 \end{array}\right]\]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:
\[\left.q\left(x_{1}, x_{2}, x_{3}\right)=\mid x_{1} x_{2} x_{3}\right]\left[\begin{array}{rrr} 2 & 0 & -1 \\ 1 & 5 & 2 \\ -2 & 1 & 1 \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]\][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:
\[q(x, y, z)=x^{2}-2 y z+x z\][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 \(P^{\prime} \mathrm{H} \overline{\mathrm{P}}\) is diagonal.
[10M]
5) Let \(L: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}\) 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 \(\mathrm{L}: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}\) 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 \(\mathrm{S}=\left[\mathrm{v}_{\mathrm{1}}, \mathrm{v}_{2}, \mathrm{v}_{3}\right]\) and \(\mathrm{T}=\left[\mathrm{w}_{1}, \mathrm{w}_{2}\right]\) be bases for \(\mathrm{R}^{3}\) and \(\mathrm{R}^{2}\) respectively, where
\[\begin{array}{l} v_1=\left[\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right], v_{2}=\left[\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right], v_{3}=\left[\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right] \\ w_1=\left[\begin{array}{ll} 1 \\ 0 \end{array}\right] \text { and } w_{2}=\left[\begin{array}{l} 0 \\ 1 \end{array}\right] \end{array}\]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 \(\mathrm{D}: \mathrm{V} \rightarrow \mathrm{V}\) be the differentiation operator defined by \(\mathrm{D}\left(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}\right)=a_{1}+2 a_{2} x+3 a_{3} x^{2}, x \in \mathrm{R}\).
(i) Show that \(\mathrm{B}\left\{1, \mathrm{x}, \mathrm{x}^{2}, \mathrm{x}^{3}\right\}\) is a basis for \(\mathrm{V}\).
(ii) Find the matrix \([D]_{\mathrm{B}}\) with respect to \(\mathrm{B}\) of \(\mathrm{D}\).
(iii) Show that \(\mathrm{B}^{\prime}\left\{1,(\mathrm{x}+1),(\mathrm{x}+1)^{2},(\mathrm{x}+1)^{3}\right\}\) is a basis for \(V\).
(iv) Find the matrix \([D]_\mathrm{B}\), with respect to \(\mathrm{B}'\) of D.
(v) Find the matrix \([\mathrm{D}]_{B',B}\) of \(\mathrm{D}\) relative to \(\mathrm{B}^{\prime}\) 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 \(S= \left\{ v, v_1, v_2, v_2, v_3 \right\}\) linearly independent?
[10M]
4) Determine a non-singular matrix \(P\) such that \(P^{\prime} A P\) is a diagonal matrix, where \(P^{\prime}\) 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 \(4P^{\prime} A P\) is a diagonal matrix, where \(P^{\prime}\) 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 \(\phi=n\left(x_{1}^{2}+x_{2}^{2}+\ldots .+x_{n}^{2}\right)-\left(x_{1}+x_{2}+\ldots+x_{n}\right)^{2}\) in \(n\) variables is positive semi-definite.
[10M]