📘 Linear Algebra

1. State the problem: This request asks for 100 descriptive questions and answers on advanced linear algebra topics suitable for UPSC mains standard. 2. Explanation: Creating such

1. Stating the problem: We need to find the cosine of the angle between vector $G = (9,0,3)$ and each film vector $F_i = (a_i,d_i,h_i)$ for $i=1$ to $5$. 2. Recall the cosine simil

1. **Problem statement:** Given the matrix $$\begin{pmatrix}

1. The problem asks to determine if $\langle P_{3}(\mathbb{R}), +, \cdot \rangle$ is a vector space. 2. $P_{3}(\mathbb{R})$ is the set of all real polynomials of degree at most 3.

1. **State the problem**: Find the eigenvalues and eigenvectors of the matrix $$A = \begin{bmatrix} 1 & 1 & -2 \\ -1 & 2 & 1 \\ 0 & 1 & -1 \end{bmatrix}.$$

1. Өгөгдсөн Крамерийн дүрмийн дагуу шуудтман тэгшитгэлийн системийг бодно: $$\begin{cases} 2x_1 + 2x_2 + 3x_3 = 5 \\ 3x_1 + 4x_2 + x_3 = 13 \\ x_1 + 2x_2 + 4x_3 = 2 \end{cases}$$

1. Find a basis and dimension for the solution space of the system: \begin{cases}

1. **State the problem:** We want to find the standard matrix of the linear transformation $$T : \mathbb{R}^3 \to \mathbb{R}^3$$ defined by $$T(x,y,z) = (x + 2y - 3z, \; 2x + y + z

1. **Problem statement:** Determine which given statements about the linear transformation $T:\mathbb{R}^2 \to \mathbb{R}^2$ defined by rotation through angle $\theta$ are false. 2

1. **Stating the problem:** We have a linear transformation $$T : \mathbb{P}_2(\mathbb{R}) \to \mathbb{P}_3(\mathbb{R})$$ defined by

1. **State the problem:** We have a linear transformation $T : \mathbb{R}^3 \to \mathbb{R}^2$ defined by $$T(x,y,z) = (x + y - z, x - y - z).$$ We want to find a basis for the kern

1. **Stating the problem:** We have a linear transformation $T : \mathbb{P}_3 \to \mathbb{P}_3$ defined by $$T(p(x)) = p'(x) + p(x),$$ where $\mathbb{P}_3$ is the space of all real

1. **Problem statement:** We are given matrices $$A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$$

1. Problem Statement: Find the reduced row echelon form (RREF) of each of the given matrices and record the row operations. 2. For matrix (a):

1. مشكلة: لدينا مصفوفة مربعة ثلاثية الأبعاد ونريد إيجاد القيم الذاتية والاشعة الذاتية لها. 2. مثال للمصفوفة:

1. **نص المشكلة:** لدينا مصفوفة مربعة ونريد إيجاد القيم الذاتية (eigenvalues) والاشعة الذاتية (eigenvectors) لها. 2. **مثال المصفوفة:** لنأخذ المصفوفة $$A=\begin{bmatrix}4 & 1\\2 &

1. Problem 1: Traffic Flow Rates We have a network with known and unknown traffic flow rates $x_1,x_2,x_3,x_4,x_5,x_6,x_7$. The goal is to:

1. **State the problem:** We are given matrix

1. **Problem 1:** Given matrices $A = [a_{ij}]$ of size $m \times m$, $B = [b_{ij}]$ of size $n \times 3$, and $C = [c_{ij}]$ of size $p \times q$ such that the products $AB$ and $

1. **Problem:** Given matrices $A = [a_{ij}]_{m\times4}$, $B = [b_{ij}]_{1\times3}$, and $C = [c_{ij}]_{p\times q}$, products $AB$ and $AC$ are both defined and are square matrices

1. **Stating the problem:** We are given vectors \(\mathbf{v}_1 = [1, 1, -1, 0]\) and \(\mathbf{v}_2 = [0, 2, 0, 1]\) that form a basis for subspace \(W\subseteq \mathbb{R}^4\). We