 # IGNOU BMTE-141 Solved Assignment 2023 | B.Sc (G) CBCS

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

101.00

Details For BMTE-141 Solved Assignment

## IGNOU BMTE-141 Assignment Question Paper 2023

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.

$$cos\:2\theta =cos^2\theta -sin^2\theta$$

## BMTE-141 Sample Solution 2023

Simply click “Install” to download and install the app, and then follow the instructions to purchase the required assignment solution. Currently, the app is only available for Android devices. We are working on making the app available for iOS in the future, but it is not currently available for iOS devices.

Yes, It is Complete Solution, a comprehensive solution to the assignments for IGNOU. Valid from January 1, 2023 to December 31, 2023.

Yes, the Complete Solution is aligned with the IGNOU requirements and has been solved accordingly.

Yes, the Complete Solution is guaranteed to be error-free.The solutions are thoroughly researched and verified by subject matter experts to ensure their accuracy.

As of now, you have access to the Complete Solution for a period of 6 months after the date of purchase, which is sufficient to complete the assignment. However, we can extend the access period upon request. You can access the solution anytime through our app.

The app provides complete solutions for all assignment questions. If you still need help, you can contact the support team for assistance at Whatsapp +91-9958288900

No, access to the educational materials is limited to one device only, where you have first logged in. Logging in on multiple devices is not allowed and may result in the revocation of access to the educational materials.

Payments can be made through various secure online payment methods available in the app.Your payment information is protected with industry-standard security measures to ensure its confidentiality and safety. You will receive a receipt for your payment through email or within the app, depending on your preference.

The instructions for formatting your assignments are detailed in the Assignment Booklet, which includes details on paper size, margins, precision, and submission requirements. It is important to strictly follow these instructions to facilitate evaluation and avoid delays.

$$cos\:2\theta =1-2\:sin^2\theta$$

## Terms and Conditions

• The educational materials provided in the app are the sole property of the app owner and are protected by copyright laws.
• Reproduction, distribution, or sale of the educational materials without prior written consent from the app owner is strictly prohibited and may result in legal consequences.
• Any attempt to modify, alter, or use the educational materials for commercial purposes is strictly prohibited.
• The app owner reserves the right to revoke access to the educational materials at any time without notice for any violation of these terms and conditions.
• The app owner is not responsible for any damages or losses resulting from the use of the educational materials.
• The app owner reserves the right to modify these terms and conditions at any time without notice.
• By accessing and using the app, you agree to abide by these terms and conditions.
• Access to the educational materials is limited to one device only. Logging in to the app on multiple devices is not allowed and may result in the revocation of access to the educational materials.

Our educational materials are solely available on our website and application only. Users and students can report the dealing or selling of the copied version of our educational materials by any third party at our email address (abstract4math@gmail.com) or mobile no. (+91-9958288900).

In return, such users/students can expect free our educational materials/assignments and other benefits as a bonafide gesture which will be completely dependent upon our discretion.

Insert math as
$${}$$