IAS PYQs 2
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 \(\leq 2\). Express \(3 t^{2}-5 t+4\) as a linear combination of \(f_{1}, f_{2}, f_{3}\).
[10M]
2) If \(T : V _{4}( R ) \rightarrow V _{3}( R )\) is a linear transformation defined by \(T ( a , b , c , d )=( a - b + c + d , a +2 c - d , a + b +3 c -3 d )\)
For \(a\), \(b\), \(d \in R\), then verify that Rank \(T +\) Nullity \(T =\operatorname{dim} V _{4}( R )\).
[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
\(a_{i i}=n, a_{i j}=r\) if \(i \neq j,\) show that
\([ A -( n - r ) H ][ A -( n - r + nr ) I ]=0\)
Where \(b_{i i}=1, b_{i j}=\rho\) when \(i \neq j\) and \(\rho\neq 1, \rho \neq \dfrac{1}{1-n}\).
[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 \(\left[\begin{array}{cccc} 0 & c & -b & a^{\prime} \\ -c & 0 & a & b^{\prime} \\ b & -a & 0 & c^{\prime} \\ -a^{\prime} & -b^{\prime} & -c^{\prime} & 0 \end{array}\right]\) where \(a a^{\prime}+b b^{\prime}+c c^{\prime}=0\) 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
\[\left(\begin{array}{cc} A^{-1} & O \\ C^{-1} B A^{-1} & C^{-1} \end{array}\right)\]where \(A\), \(C\) are non-singular matrices and hence find the inverse of
\[\left(\begin{array}{llll} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{array}\right)\][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:
\[2 x_{1}^{2}+2 x_{2}^{2}+3 x_{3}^{2}-4 x_{2} x_{3}-4 x_{3} x_{1}+2 x_{1} x_{2}\][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 \(\mathrm{T}: \mathrm{V} \rightarrow\) \(\mathrm{U}\) be a linear Map. Prove that \(\operatorname{dim} \mathrm{V}=\operatorname{dim} \mathrm{R}(\mathrm{T})+\operatorname{dim} \mathrm{N}(\mathrm{T})\).
[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) \(T: B \rightarrow B^{2}\) defined by \(T(x)=(2 x,-x)\)
(ii) \(\quad T: \underline{R}^{2} \rightarrow R^{3}\) defined by \(T(x, y)=(x y, y, x)\)
(iii) \(\quad T: B^{2} \rightarrow R^{3}\) defined by \(T(x, y)=(x+y, y, x)\)
(iv) \(\quad T: R \rightarrow B^{2}\) defined by \(T(x)=(1,-1)\)
[10M]
TBC
4) Let \(\mathrm{T}: \mathrm{M}_{21} \rightarrow \mathrm{M}_{23}\) 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 \(T\left(\dfrac{x}{y}\right)\)
[10M]
5) For what values of \(n\) do the following equations
\(\begin{array}{l}
x+y+z \leq 1 \\
x+2 y+4 z=n \\
x+4 y+10 z=n^{2}
\end{array}\)
have a solution? Solve them completely in each case.
[10M]
6) Prove that a necessary and sufficient condition of a real quadratic form \(\mathrm{X}^{\prime} \mathrm{AX}\) 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 \(\rightarrow\left[\begin{array}{ll}2 & 1 \\ 4 & 3\end{array}\right]\)
[10M]
8) Transform the following to the diagonal forms and give the transformation employed:
\[x^{2}+2 y, 8 x^{2}-4 x y+5 y^{2}\][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 \(\mathrm{T}\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}\right)=\left(\mathrm{x}_{1}-\mathrm{x}_{2}+2 \mathrm{x}_{3}, 2 \mathrm{x}_{1}+\mathrm{x}_{2^{7}}-\mathrm{x}_{1}-2 \mathrm{x}_{2}+2 \mathrm{x}_{3}\right)\).
(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 \(A^{6}-4 A^{5}+8 A^{4}-12 A^{3}+14 \mathrm{A}^{2}\) as a linear polynomial in \(\mathrm{A}\).
4) Let \(T\) be the linear transformation from \(R^{2}\) to itself defined by \(\mathrm{T}\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right)=\left(-\mathrm{x}_{2}, \mathrm{x}_{1}\right)\).
(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 \(\beta= \left\{ \alpha_1,\alpha_2 \right\}\) where \(\alpha_1=(1,2)\) and \(\alpha_2=(1,-1)\)?
5) Determine a non-singular matrix \(\mathrm{P}\) such that \(\mathrm{P}^{\mathrm{t}} \mathrm{AP}\) 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 \(\vert I-U \vert \neq 0\). 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 \(\mathrm{e}^{\mathrm{it}_{1}}, \mathrm{e}^{\mathrm{it}_{2}}, \ldots \ldots \ldots \mathrm{e}^{\mathrm{it}_{n}}\) be the eigen values of \(\mathrm{U},\) the eigen values of \(\mathrm{H}\) are \(\cot (\alpha_1 / 2), \cot (\alpha_2 / 2), \ldots \ldots \ldots\) \(\cot (\alpha_n / 2)\).
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 \(\operatorname{dim} \mathrm{V} / \mathrm{W}=\operatorname{dimV}-\operatorname{dimW}\)
3) Show that all vectors \(\left(x_{1}, x_{2}, x_{3}, x_{4}\right)\) 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 \(A=\left[\begin{array}{ll}0 & 1 \\ -2 & 3\end{array}\right]\)