📘 linear algebra

1. **State the problem:** Find the linear transformation $T: V_3(\mathbb{R}) \to V_3(\mathbb{R})$ determined by the matrix $$

1. **Problem statement:** We are given the vector space $P_2$ of polynomials of degree at most 2 with inner product $\langle p,q \rangle = \int_0^1 p(x)q(x)\,dx$. The basis is $S =

1. **Problem Statement:** Determine which of the given vectors (a) [3, 9, 0], (b) [-4, 0, -2], (c) [3, 2, 1], and (d) [3, 3, 1] are linearly dependent with the vectors \(\mathbf{v_

1. **Problem Statement:** Prove that a set of mutually orthogonal non-zero vectors is always linearly independent. 2. **Definitions and Formula:**

1. **Problem statement:** Verify that for a matrix $A$ and integer $k \geq 1$, the transpose of the power satisfies $$(A^k)^T = (A^T)^k.$$ 2. **Recall the transpose property:** For

1. **Problem Statement:** We are given the vector $\mathbf{v} = [-1, 3]$. We need to sketch this vector and find its length. 2. **Formula for Length of a Vector:** The length (or m

1. Verilmiş məsələ: $A=\begin{pmatrix} 2 & 4 \\ -1 & -3 \end{pmatrix}$ matrisi üçün məxsusi vektorları tapmaq. 2. Məxsusi vektor tapmaq üçün əvvəlcə məxsusi ədədləri (eigenvalues)

1. Problem: Find all minors and cofactors of matrix \(A = \begin{bmatrix}1 & -2 & 3 \\ 6 & 7 & -1 \\ -3 & 1 & 4\end{bmatrix}\). 2. The minor \(M_{ij}\) of an element is the determi

1. **Problem Statement:** Find the eigenvalues and eigenvectors of matrix \( A = \begin{pmatrix} 11 & 9 & -2 \\ -8 & -6 & 2 \\ 4 & 4 & 1 \end{pmatrix} \). 2. **Characteristic Equat

1. 問題陳述：給定三階矩陣 $$A=\begin{bmatrix}5 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 3 & 3\end{bmatrix}$$，求其行空間 (RS(A))、列空間 (CS(A))、核空間 (Ker(A)) 及左核空間 (LKer(A))。 2. 公式與定義：

1. **Problem:** Show that the reflection of vector $v = (1,1)$ about the y-axis produces the same result as a rotation of $v$ through 90° counterclockwise about the origin. 2. **Re

1. **Problem statement:** Given the vector $v = (1,1)$ in $\mathbb{R}^2$, show that reflecting $v$ about the y-axis produces the same result as rotating $v$ by 90° counterclockwise

1. **Problem statement:** Given a vector $v = (1,1)$ in $\mathbb{R}^2$, show that reflecting $v$ about the $y$-axis produces the same result as rotating $v$ by 90° counterclockwise

1. **State the problem:** We are given the system of linear equations:

1. The problem involves evaluating the expression $$\left| x+3 \begin{bmatrix}a&b\\c&d\end{bmatrix}^n \right|$$ where the matrix $$\begin{bmatrix}a&b\\c&d\end{bmatrix}$$ is raised

1. Statement of the problem. We are given the matrices A and B and are asked to compute $2A-3B$.

1. **Problem Statement:** Find the standard matrix $A$ representing the linear operator $L: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $$L\begin{pmatrix}x \\ y\end{pmatrix} = \begin

1. **Problem Statement:** Given the linear transformation $L(\begin{bmatrix}x \\ y\end{bmatrix}) = \begin{bmatrix}x + 2y \\ x + y\end{bmatrix}$, find the standard matrix $A$ of $L$

1. **State the problem:** Given matrices \(P\) and \(Q\), and the equation \(P + Q + R = 0\), find matrix \(R\). 2. **Recall the rule:** To find \(R\), rearrange the equation as \(

1. The user asks about finding matrix $M$ in the first prompt. 2. Since no specific matrix $M$ or context is given in this message, I cannot provide a direct calculation or formula

1. The question asks how matrix $M$ was obtained. 2. To explain this, we need to know the context or the operation that produced matrix $M$ (e.g., multiplication, transformation, o