📘 Linear Algebra

1. **Problem statement:** We have a line \(\mathcal{L}\) (from part 2, assumed parametric form \(x=1+t, y=2-2t, z=3t\)) and a plane \(\mathcal{P}\) given by the equation \(x + 3y -

1. **Stating the problem:** We are given matrices \(A = \begin{pmatrix} 2 & 1 \ \end{pmatrix}\) and \(B = \begin{pmatrix} 5 & 7 \end{pmatrix}\).

1. **Problem statement:** Check if the matrix $$A = \begin{bmatrix}1 & 1 & 1 \\ 1 & 1 & -1 \\ 1 & -1 & -1\end{bmatrix}$$

1. The problem is to understand the matrix \(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\) and its properties. 2. This is a 2x2 matrix with elements \(a, b, c, d\).

1. **Problem Statement:** (a) Check if the matrix $$A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & -1 \\ 1 & -1 & -1 \end{bmatrix}$$ is diagonalizable. If yes, find diagonal matrix $$D$$

1. **Problem:** Determine the truth value of each statement and justify with proof or counter-example. 2. **(a) Eigenvalues of a symmetric matrix are real.**

1. **Problem (d):** Check if the vector $\mathbf{v} = \begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix}$ is an eigenvector of the matrix $$A = \begin{bmatrix}2 & -1 & 0 \\ 0 & 2 & -1 \\ 2 &

1. **Problem Statement:** Given subspaces

1. The problem is to find the value of \(b_{ii}\), which typically refers to the diagonal element in the second row and second column of a matrix \(B\). 2. To solve this, we need t

1. The problem is to create a matrix from the given modified variable names and propose a method to solve it. 2. Since no specific variables or equations are provided, let's assume

1. **State the problem:** We have a system of four linear equations:

1. **Problem statement:** We have a trade matrix $$A = \begin{pmatrix} 0.15 & 0.5 & 0.25 \\ 0.35 & 0.3 & 0.45 \\ 0.5 & 0.2 & 0.3 \end{pmatrix}$$ and budgets vector $$\vec{x} = \beg

1. **Problem statement:** Show that $X=\mathbb{R}^2$ is a normed linear space with a given norm. 2. **Definition of normed linear space:** A normed linear space is a vector space $

1. **Problem Statement:** Show that $X=\mathbb{R}^2$ is a normed linear space with a given norm. 2. **Definition:** A normed linear space is a vector space $X$ over $\mathbb{R}$ or

1. **State the problem:** Find the eigenvalues and eigenvectors of the matrix $$A = \begin{bmatrix} 14 & -10 \\ 5 & -1 \end{bmatrix}$$. 2. **Eigenvalues formula:** Eigenvalues $\la

1. **State the problem:** Find bases for the row space, column space, and null space of the matrix $$ A = \begin{bmatrix} 1 & 2 & -3 & 4 & 0 \\ -1 & 0 & 2 & -4 & 1 \\ -1 & 4 & 0 &

1. The problem is to understand the three types of pivot operations used in matrix row operations. 2. The three pivot operations are:

1. **Problem statement:** Determine for which values of $k \in \mathbb{R}$ the matrix $$

1. **Problem Statement:** Find all values of $k$ for which the augmented matrix \(\begin{bmatrix} 1 & k & -4 \\ 4 & 8 & 2 \end{bmatrix}\) corresponds to a consistent linear system.

1. **Problem Statement:** Given matrix $$A = \begin{bmatrix}-7 & -9 \\ 6 & 8\end{bmatrix}$$ and its diagonalization $$A = P D P^{-1}$$, express $$A^5$$ in terms of $$P$$, a power o

1. **State the problem:** Find the transpose of the matrix $$A = \begin{bmatrix} 1 & 2 & 1 \\ -3 & -2 & 9 \\ -5 & 7 & -3 \end{bmatrix}$$