Course Code: BMTE-141

Assignment Code: BMTE-141/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 \(W=\left\{(x, y, z) \in \mathbb{R}^{3} \mid x+y-z=0\right\}\) 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 \(T(x, y)=(2 x+3 y, 2 x-3 y)\).

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 \(T(x, y)=(2 y, x-y)\) 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 Cayley-Hamilton 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{Part-B (40 Marks)}

1) a) Let \(S\) be any non-empty 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 \(B=\left\{1,2 x+1,(x-1)^{2}\right\}\) 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:

\[
\begin{array}{r}
x+2 y+z=3 \\
2 x-y+2 z=1 \\
3 x+y+z=0
\end{array}
\]

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 self-adjoint.

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 non-zero, 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.