Math Problem Solver

Here is the typed version of the questions from the uploaded paper:

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**DEPARTMENT OF BIOMEDICAL ENGINEERING**

**UNIVERSITY OF ENGINEERING AND TECHNOLOGY, NAROWAL CAMPUS**

**MA: Linear Algebra**

**Semester: 3rd**

**Examination: Final Term**

**Session: 202X**

**Date: Jan 3, 2024**

**Time Allowed: 02 Hours**

**Total Marks: 40**

**Note: Attempt all questions.**

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### **Q1:**

(a) Let \( u, v \in \mathbb{R}^3 \), where \( u = (x_1, x_2, x_3) \) and \( v = (y_1, y_2, y_3) \). Then show that:

\[(u \cdot v) = x_1y_1 + x_2y_2 + x_3y_3\]

is an inner product space.

(b) Define Orthogonal Matrix. Show that the columns of matrix \( A \) form an orthonormal set, where:

\[A = \begin{bmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix}\]

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### **Q2:**

(a) Given a square matrix \( A = \begin{bmatrix} 4 & -1 \\ -1 & 4 \end{bmatrix} \):

(i) Find the characteristic polynomial, eigenvalues, and construct eigenvectors corresponding to each eigenvalue.

(ii) Express matrix \( A \) in Jordan Canonical form.

(b) If \( A = \begin{bmatrix} 4 & -3 \\ -3 & 4 \end{bmatrix} \) and \( P = \begin{bmatrix} 3 & 4 \\ 4 & -3 \end{bmatrix} \), then construct a matrix \( D \) that is similar to \( A \) by similarity transformation.

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### **Q3:**

(a) Write down the properties for a transformation to be linear. Define a mapping \( T: \mathbb{R}^2 \to \mathbb{R}^2 \), verify whether \( T \) is a linear transformation:

\[T(x, y) = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}\]

(b) Find the inverse linear transformation. Show the detail of your work, where:

\[y_1 = 3x_1 + 2x_2, \quad y_2 = 4x_1 + x_2\]

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Let me know if you need detailed solutions to any part!