IAS PYQs 3
1988
1) Show that a linear transformation of a vector space \(\mathrm{V}_{\mathrm{m}}\) of dimension \(\mathrm{m}\) into a vector space \(\mathrm{V}_{\mathrm{n}}\) of dimension \(n\) over the same field can be represented by a matrix. \(\mathrm{T}\) is linear transformation of \(\mathrm{v}_{2}\) into \(\mathrm{v}_{4}\) such that \(\mathrm{T}(3,1)=(4,1,2,1)\), \(\mathrm{T}(-1,2)=(3,0,-2,1)\) find the matrix of \(\mathrm{T}\).
2) Determine a basis of the subspace spanned by the vectors \(v_{1}=(1,2,3)\), \(v_{2}=(2,1,-1)\), \(v_{3}=(1,-1,-4)\), \(v_{4}=(4,2,-2)\).
3) Show that it is impossible for any \(2 \times 2\) real symmetric matrix of the form \(\left[\begin{array}{ll}a_{1} & b \\ b & a_{2}\end{array}\right](b \neq 0)\) to have identical eigen values.
4) Prove that the eigen values of a Hermitian matrix are all real and that an eigen value of a skew Hermitian matrix is either zero or a pure imaginary number.
5) By converting \(A\) to an echelon matrix, determine the rank of
\[A=\left[\begin{array}{llllll}0 & 0 & 1 & 2 & 8 & 9 \\ 0 & 0 & 4 & 6 & 5 & 3 \\ 0 & 2 & 3 & 1 & 4 & 7 \\ 0 & 3 & 0 & 9 & 3 & 7 \\ 0 & 0 & 5 & 7 & 3 & 1\end{array}\right]\]6) Given \(\mathrm{AB}=\mathrm{AC}\) does it follow that \(\mathrm{B}=\mathrm{C}\)? Can you provide a counter example?
7) Find a non- singular transformation of three variables which simultaneously diagonalizes.
\[A=\left[\begin{array}{ccc} 0 & -1 & 0 \\ -1 & -1 & 1 \\ 0 & 1 & 0 \end{array}\right]\] \[B=\left[\begin{array}{ccc} 2 & 1 & -2 \\ 1 & 2 & -2 \\ -2 & -2 & 3 \end{array}\right]\]Give the diagonal form of \(\mathrm{A}\).
1987
1.(i) Find all matrices which commute with the matrix \(\left[\begin{array}{rr}7 & -3 \\ 5 & -2\end{array}\right]\)
1.(ii) Prove that the product of two \(\mathrm{n} \times \mathrm{n}\) symmetric matrices is a symmetric matrix, if and only if the matrices commute.
2) Show that the rank of the product of two square matrices \(A, B\) each of order \(n\), satisfies the inequality \(r_{A}+r_{n}-n \leq r_{A B} \leq \min \left(r_{A}, r_{B}\right)\), where \(\mathrm{r}_{0}\) stands for the rank of a square matrix \(\mathrm{C}\).
3) If \(1 " a^{\prime \prime} 5,\) find the rank of the matrix \(\left(\begin{array}{cccc}1 & 1 & 1 & 1 \\ 1 & 3 & -2 & \mathrm{a} \\ 2 & 2 \mathrm{a}-2 & -\mathrm{a}-2 & 3 \mathrm{a}-1 \\ 3 & \mathrm{a}+2 & -3 & 2 \mathrm{a}+1\end{array}\right)\)
TBC
4) If the eigen values of a matrix A are \(\lambda_j, \mathrm{j}=1,2 \ldots . . \mathrm{n}\) and if \(f(x)\) is a polynomial in \(x,\) show that the eigen values of the matrix, \(\mathrm{f}(\mathrm{A})\) are \(\mathrm{f}(\lambda_j), \mathrm{j}=1,2 \ldots \ldots \ldots \mathrm{n}\).
5) If \(\mathrm{A}\) is a skew symmetric matrix and \(\mathrm{I},\) the corresponding unit matrix, show that \((I-A)(I+A)^{-1}\) is orthogonal. Hence construct an orthogonal matrix, if \(\mathrm{A}=\left(\begin{array}{cc}0 & \mathrm{a} / \mathrm{b} \\ -\mathrm{a} / \mathrm{b} & 0\end{array}\right)\)
6) (i) If \(\mathrm{A}, \mathrm{B}\) are two arbitrary square matrices of which \(A\) is non-singular, show that \(AB\) and \(BA\) have the same characteristic polynomial.
(ii) Show that a real matrix \(\mathrm{A}\) is orthogonal, if and only if \(\vert Ax \vert = \vert x \vert\) for all real vectors \(x\).
7) Show that a necessary and sufficient condition for a system of Linear equations to be consistent is that the rank of the coefficient matrix is equal to the rank of the augmented matrix. Hence show that the system \(x+2 y+5 z+9=0\), \(x-y+3 z-2=0\), \(3 x-6 y-z-25=0\) is consistent and has a unique solution. Determine this solution.
8) In an \(\mathrm{n}\)-dimensional vector space the system of vectors \(a_{j}, j=1,2\)\(\mathrm{r}\) are linearly independent and can be linearly expressed to terms of the vectors \(\beta_{k}, \mathrm{k}=1,2, \ldots \ldots, \mathrm{s}\). show that \(r^{\prime \prime} s\).
Find a maximal linearly independent subsystem of the system of linear forms \(f_{1}=x+2 y+z+3 t\), \(f_{2}=4 x-y-5 z-6 t\), \(f_{3}=x-3 y-4 z-7 t\), \(f_{4}=2 x+y-z\).
9) Let \(T\): \(V \rightarrow W\) be a linear transformation. If \(\mathrm{V}\) is finite dimensional then, show that rank (T) + nullity \((\mathrm{T})\)=\(\operatorname{dim} \mathrm{V}\).
10) Prove that two finite dimensional vector spaces \(\mathrm{V}\) and \(\mathrm{W}\) over the same field \(\mathrm{F}\) are isomorphic if they are of the same dimension \(\mathrm{n}\).
11.(a) Prove that every square matrix is a root of its characteristic polynomial.
11.(b) Determine the eigen values and the corresponding eigen vectors of \(A=\left[\begin{array}{lll}2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2\end{array}\right]\)
12) Let \(\mathrm{A}\) and \(\mathrm{B}\) be \(\mathrm{n}\) - square matrices over \(\mathrm{F}\). show that \(\mathrm{AB}\) and \(\mathrm{BA}\) have the same eigen values.
1986
1) If \(\mathrm{A}\), \(\mathrm{B}\), \(\mathrm{C}\) are three \(\mathrm{n} \times \mathrm{n}\) matrices, show that \(\mathrm{A}(\mathrm{BC})\)=\((\mathrm{AB}) \mathrm{C}\). Show by an example that the matrix multiplication is non-commutative.
2) Examine the correctness or otherwise of the statements:
(i) The division law is not valid in matrix algebra.
(ii) If \(\mathrm{A}\), \(\mathrm{B}\) are square matrices each of order \('n'\) and \(I\) is the corresonding unit matrix, the equation \(\mathrm{AB}-\mathrm{BA}\)=\(\mathrm{I}\) can never hold.
3) Find a \(3 \times 3\) matrix \(\mathrm{X}\) such that
\(\left[\begin{array}{ccc}1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1\end{array}\right] \mathrm{X}=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & 1\end{array}\right]\) may hold.
4) Find a maximal linearly independent subsystem in the system of vectors:
\(\mathrm{V}_{1}=(2,-2,-4)\), \(\mathrm{V}_{2}=(1,9,3)\), \(\mathrm{V}_{3}=(-2,-4,1)\) and \(\mathrm{V}_{4}=(3,7,-1)\).
5) Show that the system of equations
\(4 x+y-2 z+w=3\)
\(x-2 y-z+2 w=2\)
\(2 x+5 y-w=-1\)
\(3 x+3 y-z-3 w=1\)
although consistent, is not uniquely solvable. Determine the general solution by choosing \(\mathrm{x}\) as a parameter.
6) Show that
(i) a square matrix is singular, iff, atleast one of its characteristic roots (i.e., eigen values) is zero.
(ii) the rank of an \(n \times n\) matrix A remains unchanged, if \(\mathrm{A}\) is pre-multiplied or post-multiplied by a non-singular \(\mathrm{n} \times \mathrm{n}\) matrix \(\mathrm{X}\) and that rank \(\left(X^{-1} A X\right)=\mathrm{rank}\) A. Prove that \(\mathrm{V}\) is a vector space over \(\mathrm{K}\).
7) Find all the eigenvalues and a basis for each eigen space for the matrix \(\left[\begin{array}{rrr}1 & -3 & 3 \\ 3 & -5 & 3 \\ 6 & -6 & 4\end{array}\right]\).
8) Prove that every square matrix satisfies its own characteristic equation.
9) Prove that the rank of the product of two matrices cannot exceed the rank of either of them.
10) If \(\mathrm{M}\), \(\mathrm{N}\) are two subspaces of a vector space \(\mathrm{S}\), show that their dimensions satisfy.
\[\operatorname{dim}(M)+\operatorname{dim}(N)=\operatorname{dim}(M \cap N)+\operatorname{dim}(M+N)\]11) Let \(\mathrm{V}\) and \(\mathrm{W}\) be vector spaces over the same field \(\mathrm{F}\) and \(\mathrm{dimV}=\mathrm{n}\), let \(\left\{\mathrm{e}_{1}, \mathrm{e}_{2}, \ldots \ldots, \mathrm{e}_{n}\right\}\) be a basis of \(V\). Show that a map \(f\): \(\left\{e_{1}, e_{2}, \ldots \ldots, e_{0}\right\} \rightarrow W\) can be uniquely extended to a linear transformation \(T\): \(V \rightarrow W\), whose restriction to the given basis \(\{\mathrm{e}\}\) is \(\mathrm{f}\), that is \(\mathrm{T}\left(\mathrm{e}_{\mathrm{j}}\right)=\mathrm{f}(\mathrm{e})\); \(\mathrm{j}=1,2, \ldots \ldots \ldots \ldots, \mathrm{n}\).
12.(i) If \(\mathrm{A}\) and \(\mathrm{B}\) are two linear transformations on the space, and if \(\mathrm{A}^{-1}\) and \(\mathrm{B}^{-1}\) exist, then show that \((A B)^{-1}\) exists and \((A B)^{-1}=B^{-1} A^{-1}\).
12.(ii) Prove that similar matrices have the same characteristic polynomial and hence the same eigen values.
12.(iii) Prove that the eigen values of a Hermitian matrix are real.
13) Reduce \(2 x^{2}+4 x y+5 y^{2}+4 x+13 y-1 / 2=0\) to canonical form
(i) Find reciprocal of the matrix \(T=\left[\begin{array}{lll} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right]\)
, show that the transform of the matrix \(A=\dfrac{1}{2}\left[\begin{array}{lll}b+c & c-a & b-a \\ c-b & c+a & a-b \\ b-c & a-c & a+b\end{array}\right]\) by \(\mathrm{T}\), i.e. TAT is the diagonal matrix. Determine the eigen values of the matrix \(A\).
1985
1) Let \(M_{1}=\left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right), M_{2}=\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right), M_{3}=\left(\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right)\), \(M_{4}=\left(\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right)\) prove that the set \(\left\{\mathrm{M}_{1}, \mathrm{M}_{2}, \mathrm{M}_{3}, \mathrm{M}_{4}\right\}\) form the basis of the vector space of \(2 \times 2\) matrices.
2) Find the inverse of the matrix \(\left(\begin{array}{lll}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{array}\right)\).
3) Consider the basis \(\mathrm{S}=\left\{\mathrm{V}_{1}, \mathrm{V}_{2}, \mathrm{V}_{3}\right\}\) of \(R^{3}\), where \(\mathrm{V}_{1}=(1,1,1)\), \(\mathrm{V}_{2}=(1,1,0)\), \(\mathrm{V}_{3}=(1,0,0)\). Express \((2,-3,5)\) in terms of the basis \(\mathrm{V}_{1}\), \(\mathrm{V}_{2}\), \(\mathrm{V}_{3}\). Let \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}\) be defined as \(\mathrm{T}\left(\mathrm{V}_{1}\right)=(1,0)\), \(T\left(V_{2}\right)=(2,-1)\), \(T\left(V_{3}\right)=(4,3) \cdot\). Find \(\mathrm{T}(2,-3,5)\).
4) Reduce the matrix to echelon form \(\left[\begin{array}{ccc}6 & 3 & -4 \\ -4 & 1 & -6 \\ 1 & 2 & -5\end{array}\right]\).
5) Show that if \(\lambda\) is an eigen value of matrix \(A\) then \(\lambda^{n}\) is an eigen value of \(\mathrm{A}^{n}\) if \(\mathrm{n}\) is a positive integer.
6) Determine the vectors \((1,-2,1)\), \((2,1,-1)\), \((7,-4,1)\) in \(\mathbb{R}^{3}\) are linearly independent.
7) Solve
\(2 x_{1}+3 x_{2}+x_{3}=9\)
\(x_{1}+2 x_{2}+3 x_{3}=6\)
\(3 x_{1}+x_{2}-2 x_{3}=8\)
8) Let \(\mathrm{V}\) be the vector space of all functions from \(R\) in to \(R\); let \(v_{e}\) be the subset of even functions \(f\), \(f(-x)=f(x)\), and \(v_{0}\) be the subset of odd functions \(f\) \(f(-x)=-f(x)\). Prove that
(i) \(v_{e}\) and \(v_{0}\) are subspaces of \(v\)
(ii) \(v_{e}+v_{0}=v\)
(iii) \(v_{e} \cap v_{0}=\{0\}\)
9) Find the dimension and a basis of the solution space \(S\) of the system
\(x_{1}+2 x_{2}+2 x_{3}-x_{4}+3 x_{5}=0\)
\(x_{1}+2 x_{2}+3 x_{3}+x_{4}+x_{5}=0\)
\(3 x_{1}+6 x_{2}+8 x_{3}+x_{4}+5 x_{5}=0\)
10) Let \(w_{1}\) and \(w_{2}\) be subspaces of a finite dimensional vector space \(\mathrm{V}\). Prove that \(\left(w_{1}+w_{2}\right)^{0}=w_{1}^{0} \cap w_{2}^{0}\).
11) Prove that every matrix is a zero of its characteristic polynomial.
12) If \(\mathrm{A}\) is an orthogonal matrix and if \(\mathrm{B}=\mathrm{AP},\) where \(\mathrm{P}\) is non-singular, then prove that \(\mathrm{PB}^{-1}\) is orthogonal.
1984
1) If \(w_{1}\) and \(w_{2}\) are finite-dimensional subspaces of a vector space \(v\) then show that \(w_{1}+w_{2}\) is finite dimensional and
\[\operatorname{dim} w_{1}+\operatorname{dim} w_{2}=\operatorname{dim}\left(w_{1} \cap w_{2}\right)+\operatorname{dim}\left(w_{1}+w_{2}\right)\]2) Show that row equivalent matrices have the same rank.
3) A linear transformation \(T\) of a vector space \(\mathrm{V}\) with finite basis \(\left\{\alpha_{1}, \alpha_{2}, \ldots \alpha_{n}\right\}\) is non-singular iff the vectors \(\alpha_{1} T, \ldots \ldots \ldots . . \alpha_{n} T\) are linearly independent in \(\mathrm{V}\). When this is the case, show that \(\mathrm{T}\) has a linear inverse, \(\mathrm{T}^{-1},\) with \(\mathrm{TT}^{-1}=\mathrm{T}^{-1} \mathrm{~T}=\mathrm{I}\).
4) Solve the following system of equations:
\(3 x_{1}+2 x_{2}+2 x_{3}-5 x_{4}=8\)
\(2 x_{1}+5 x_{2}+5 x_{3}-18 x_{4}=9\)
\(4 x_{1}-x_{2}-x_{3}+8 x_{4}=7\)
5) Let \(A\) be a square matrix and \(\mathrm{T}\) be non-singular. Let \(\tilde{\mathrm{A}}=\mathrm{T}^{-1} \mathrm{AT}\). Show that
(i) \(\mathrm{A}\) and \(\tilde{\mathrm{A}}\) have the same eigen values
(ii) Trace \(\mathrm{A}=\mathrm{T}\) trace \(\tilde{\mathrm{A}}\)
(iii) If \(\mathrm{X}\) be an eigen vector of A corresponding to an eigen value then \(\mathrm{T}^{-1} \mathrm{X}\) is a eigen vector of \(\tilde{\mathrm{A}}\) corresponding to the same eigen value.
6) A \(3 \times 3\) matrix has the eigen values \(\lambda_{1}=6\), \(\lambda_{2}=2\), and \(\lambda_{3}=-1\) and the corresponding eigen vectors are \(\left(\begin{array}{c}2 \\ 3 \\ -2\end{array}\right)\), \(\left(\begin{array}{c}9 \\ 5 \\ 4\end{array}\right)\), \(\left(\begin{array}{c}4 \\ 4 \\ -1\end{array}\right)\). Find the matrix.
7) Let \(\mathrm{V}\) be the set of all functions from a non-empty set \(\mathrm{X}\) into a field \(\mathrm{K},\) for any functions \(\mathrm{f}, \mathrm{g} \in \mathrm{v}\) and any scalar \(\mathrm{k} \in \mathrm{K},\) let \((\mathrm{f}+\mathrm{g})\) and \(\mathrm{kf}\) be the functions in \(v\) defined as follows:
\((f+g)(x)=f(x)+g(x),(k f)(x)=k f(x)\) for every \(x \in X\).
8) Find all the eigenvalues and a basis for each eigen space for the matrix \(\left(\begin{array}{ccc}1 & -3 & 3 \\ 3 & -5 & 3 \\ 6 & -6 & 4\end{array}\right)\).
9) Prove that every square matrix satisfies its own characteristic equation.
10) Prove that the rank of the product of two matrices cannot exceed the rank of either of them.
1983
1) Let \(\mathrm{V}\) be a finitely generated vector space. Show that \(\mathrm{V}\) has a finite basis and any two bases of \(\mathrm{V}\) have the same number of vectors.
2) Let \(\mathrm{V}\) be the vector space of polynomials of degree \(\leq3\). Determine whether the following vectors of \(\mathrm{V}\) are linearly dependent or independent: \(u=t^{3}-3 t^{2}+5 t+1\), \(v=t^{3}-t^{2}+8 t+2\), \(w=2 t^{3}-4 t^{2}+9 t+5\).
3) For any linear transformation \(T: V_{1} \rightarrow V_{2}\), prove that \(rank (T)\)= \(\min \left(\operatorname{dim} V_{1}, \operatorname{dim} V_{2}\right)\).
4) Show that every non-singular matrix can be expressed as a product of elementary matrices.
5) Reduce the matrix \(\left[\begin{array}{llll}0 & 1 & 2 & 1 \\ 1 & 2 & 3 & 2 \\ 3 & 1 & 1 & 3\end{array}\right]\) to its normal form and hence or otherwise determine its rank.
6) Show that the equations
\(x+y+z=3\)
\(3 x-5 y+2 z=8\)
\(5 x-3 y+4 z=14\) are consistent and solve them.
7) Prove that every square matrix satisfies its characteristic equation. Use this result to find the inverse of \(\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\right]\).
8) Find the eigen values and eigen vectors of the matrix \(\left[\begin{array}{ccc}8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3\end{array}\right]\).
9) Show that the characteristic roots of an upper or lower triangular matrix are just the diagonal elements of the matrix.