IGNOU BMTE141 Solved Assignment 2023  B.Sc (G) CBCS
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IGNOU BMTE141 Assignment Question Paper 2023
Course Code: BMTE141
Assignment Code: BMTE141/TMA/2023
Maximum Marks: 100
\section{PART – A (30 Marks)}
1) i) Find the angle between the vectors \(\sqrt{2} \mathbf{i}+2 \mathbf{j}+2 \mathbf{k}\) and \(\mathbf{i}+\sqrt{2} \mathbf{j}+\sqrt{2} \mathbf{k}\).
ii) Find the vector equation of the plane determined by the points \((1,0,1),(0,1,1)\) and \((1,1,0)\).
iii) Check whether \(W=\left\{(x, y, z) \in \mathbb{R}^{3} \mid x+yz=0\right\}\) is a subspace of \(\mathbb{R}^{3}\).
iv) Check whether the set of vectors \(\left\{1+x, x+x^{2}, 1+x^{3}\right\}\) is a linearly independent set of vectors in \(\mathbf{P}_{3}\), the vector space of polynomials of degree \(\leq 3\).
v) Check whether \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\), defined by \(T(x, y)=(y, x)\) is a linear transformation.
vi) If \(\left\{v_{1}, v_{2}\right\}\) is an ordered basis of \(\mathbb{R}^{2}\) and \(\left\{f_{1}(v), f_{2}(v)\right\}\) is the corresponding dual basis find \(f_{1}\left(2 v_{1}+v_{2}\right)\) and \(f_{2}\left(v_{1}2 v_{2}\right)\).
vii) Find the kernel of the linear transformation \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) defined by \(T(x, y)=(2 x+3 y, 2 x3 y)\).
viii) Describe the linear transformation \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) such that
\[
[T]_{B}=\left[\begin{array}{ll}
1 & 2 \\
2 & 0
\end{array}\right]
\]
where \(B\) is the standard basis of \(\mathbb{R}^{2}\).
ix) Find the matrix of the linear transformation \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) defined by \(T(x, y)=(2 y, xy)\) with respect to the ordered basis \(\{(0,1),(1,0)\}\).
x) Let \(A\) be a \(2 \times 3\) matrix, \(B\) be a \(3 \times 4\) matrix and \(C\) be a \(3 \times 2\) matrix and \(D\) be a \(3 \times 4\) matrix. Is \(A B+C^{t} D\) defined? Justify your answer.
xi) Verify CayleyHamilton theorem for the matrix \(A=\left[\begin{array}{cc}1 & 1 \\ 0 & 2\end{array}\right]\).
xii) Check whether \(\left[\begin{array}{l}1 \\ 1 \\ 1 \\ 1\end{array}\right]\) is an eigenvector for the matrix \(\left[\begin{array}{llll}1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1\end{array}\right]\). What is the corresponding eigenvalue?
xiii) Let \(C[0,1]\) be the inner product space of continous real valued functions on the interval \([0,1]\) with the inner product
\[
\langle f, g\rangle=\int_{0}^{1} f(t) g(t) d t
\]
Find the inner product of the functions \(f(t)=2 t, g(t)=\frac{1}{t^{2}+5}\).
xiv) Find adjoint of the linear operator \(T: \mathbb{C}^{2} \rightarrow \mathbb{C}^{2}\) defined by \(T\left(z_{1}, z_{2}\right)=\left(z_{2}, z_{1}+i z_{2}\right)\) with respect to the standard inner product on \(\mathbb{C}^{2}\). \(\mathrm{xv})\) Find the signature of the quadratic form \(x_{1}^{2}2 x_{2}^{2}+3 x_{3}^{2}\)
\section{PartB (40 Marks)}
1) a) Let \(S\) be any nonempty set and let \(V(S)\) be the set of all real valued functions on \(\mathbb{R}\). Define addition on \(V(s)\) by \((f+g)(x)=f(x)+g(x)\) and scalar multiplication by
\((\alpha \cdot f)(x)=\alpha f(x)\). Check that \((V(S),+, \cdot)\) is a vector space.
b) Check that \(B=\left\{1,2 x+1,(x1)^{2}\right\}\) is a basis for \(\mathbf{P}_{2}\), the vector space of polynomials with real coefficients of degree \(\leq 2\)
2) a) Let \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) be a linear operator and suppose the matrix of the operator with respect to the ordered basis
\[
\begin{gathered}
B=\left\{\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right],\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right],\left[\begin{array}{l}
0 \\
1 \\
0
\end{array}\right]\right\} \\
\text { is }\left[\begin{array}{ccc}
1 & 0 & 1 \\
0 & 1 & 1 \\
1 & 0 & 1
\end{array}\right] \text {. Find the matrix of the linear transformation with respect to the basis } \\
B^{\prime}=\left\{\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right],\left[\begin{array}{c}
1 \\
0 \\
1
\end{array}\right],\left[\begin{array}{l}
0 \\
1 \\
0
\end{array}\right]\right\}
\end{gathered}
\]
b) Show that \(W=\left\{(x, 4 x, 3 x) \in \mathbb{R}^{2} \mid x \in \mathbb{R}\right\}\) is a subspace of \(\mathbb{R}^{3}\). Also find a basis for subspace \(U\) of \(\mathbb{R}^{3}\) which satisfies \(W \oplus U=\mathbb{R}^{3}\).
3) a) Find the eigenvalues and eigenvectors of the matrix \(B=\left[\begin{array}{rrr}1 & 1 & 0 \\ 1 & 3 & 0 \\ 1 & 1 & 1\end{array}\right]\). Is the matrix diagonalisable? Justify your answer.
b) Find \(\operatorname{Adj}(A)\) where \(A=\left[\begin{array}{ccc}3 & 2 & 2 \\ 1 & 1 & 0 \\ 3 & 0 & 1\end{array}\right]\). Hence find \(A^{1}\).
4) a) Solve the folowing set of simultaneous equations using Cramer’s rule:
\[
\begin{array}{r}
x+2 y+z=3 \\
2 xy+2 z=1 \\
3 x+y+z=0
\end{array}
\]
b) Find the minimal polynomial of the matrix
\[
\left[\begin{array}{rrrr}
2 & 1 & 0 & 1 \\
1 & 0 & 0 & 1 \\
2 & 2 & 1 & 3 \\
0 & 0 & 0 & 1
\end{array}\right]
\]
\section{Part C (30 marks)}
1) a) Let \(V\) be the vector space of all real valued functions that are twice differentiable in \(\mathbb{R}\) and
\[
S=\{\cos x, \sin x, x \cos x, x \sin x\}
\]
Check that \(S\) is a linearly independent set over \(\mathbb{R}\). (Hint: Consider the equation
\[
a_{0} \cos x+a_{1} \sin x+a_{2} x \cos x+a_{3} x \sin x
\]
(Put \(x=0, \pi, \frac{\pi}{2}, \frac{\pi}{4}\), etc. and find \(a_{i}\).)
b) Consider the linear operator \(T: \mathbb{C}^{3} \rightarrow \mathbb{C}^{3}\), defined by
\[
T\left(z_{1}, z_{2}, z_{3}\right)=\left(z_{1}i z_{2}, i z_{1}2 z_{2}+i z_{3},i z_{2}+z_{3}\right) \text {. }
\]
i) Compute \(T^{*}\) and check whether \(T\) is selfadjoint.
ii) Check whether \(T\) is unitary.
2) a) Let \(\left(x_{1}, x_{2}, x_{3}\right)\) and \(\left(y_{1}, y_{2}, y_{3}\right)\) represent the coordinates with respect to the bases \(B_{1}=\{(1,0,0),(1,1,0),(0,0,1)\}, B_{2}=\{(1,0,0),(0,1,1),(0,0,1)\}\). If
\[
Q(X)=x_{1}^{2}4 x_{1} x_{2}+2 x_{2} x_{3}+x_{2}^{2}+x_{3}^{2}
\]
find the representation of \(Q\) in terms of \(\left(y_{1}, y_{2}, y_{3}\right)\)
b) Find the orthogonal canonical reduction of the quadratic form \(x^{2}+y^{2}+z^{2}+4 x y+4 x z\). Also, find its principal axes.
3) Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
i) If \(W_{1}\) and \(W_{2}\) are proper subspaces of a nonzero, finite dimensional, vector space \(V\) and \(\operatorname{dim}\left(W_{1}\right)>\frac{\operatorname{dim}(V)}{2}, \operatorname{dim}\left(W_{2}\right)>\frac{\operatorname{dim}(V)}{2}\), the \(W_{1} \cap W_{2} \neq\{0\}\).
ii) If \(V\) is a vector space and \(S=\left\{v_{1}, v_{2}, \ldots, v_{n}\right\} \subset V, n \geq 3\), is such that \(v_{i} \neq v_{j}\) if \(i \neq j\), then \(S\) is a linearly independent set.
iii) If \(T_{1}, T_{2}: V \rightarrow V\) are linear operators on a finite dimensional vector space \(V\) and \(T_{1} \circ T_{2}\) is invertible, \(T_{2} \circ T_{1}\) is also invertible.
iv) If an \(n \times n\) square matrix, \(n \geq 2\) is diagonalisable then it has the same minimal polynomial and characteristic polynomial.
v) If \(T_{1}, T_{2}: V \rightarrow V\) are self adjoint operators on a finite dimensional inner product space \(V\), then \(T_{1}+T_{2}\) is also a self adjoint operator.
BMTE141 Sample Solution 2023
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