📘 linear algebra

1. **Problem statement:** Determine if $(\mathbb{R}^2, +, \cdot)$ with addition defined by $(a,b)+(c,d)=(a+c,b+d)$ and scalar multiplication defined by $\lambda \cdot (a,b) = (\lam

1. The problem is to understand the theorem: Every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix. 2. Recall definitions:

1. **State the problem:** We have a subspace $V = \text{L}\{(1,1,k)\}$ where $k$ is any real number. We want to find the dimension of $V$. 2. **Understand the subspace:** The set $

1. Problem: Determine which vectors are linear combinations of \(u = (0, -2, 2)\) and \(v = (1, 3, -1)\). Formula: A vector \(w\) is a linear combination of \(u\) and \(v\) if ther

1. **Problem:** Determine the rank of the matrix $$A = \begin{pmatrix} 0 & 0 & 2 & 2 & 0 \\ 1 & 3 & 2 & 4 & 1 \\ 2 & 6 & 2 & 6 & 2 \\ 3 & 9 & 1 & 10 & 6 \end{pmatrix}$$

1. **Problem Statement:** Determine whether the given pairs of vectors are linearly dependent. 2. **Definition:** Two vectors $\mathbf{u}$ and $\mathbf{v}$ are linearly dependent i

1. **State the problem:** Reduce the quadratic form $$2x_1^2 + x_2^2 + x_3^2 + 2x_1x_2 - 2x_1x_3 - 4x_2x_3$$ to its canonical form using an orthogonal transformation. 2. **Write th

1. The problem is to convert a matrix into its standard form. 2. Standard form for a matrix usually means putting it into row echelon form or reduced row echelon form using element

1. **Problem:** Let $T: \mathbb{R}^3 \to \mathbb{R}^2$ be defined by $T(x,y,z) = (2x - y, 3z)$. Check if $T$ is linear, find $N(T)$ and $R(T)$, and verify the Dimension theorem. 2.

1. **Problem statement:** A university server handles three types of tasks: database queries ($x$), file uploads ($y$), and API requests ($z$). The total number of tasks processed

1. The problem is to understand what a vector space is in mathematics. 2. A vector space is a collection of objects called vectors, which can be added together and multiplied by sc

1. **State the problem:** Find the adjoint (also called adjugate) of the matrices $$A = \begin{bmatrix} 3 & 5 \\ 3 & 4 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 2 &

1. **State the problem:** Find the determinant of the 4x4 matrix $$A=\begin{pmatrix}4 & -3 & 6 & -4 \\ 4 & -5 & -1 & -5 \\ -1 & -6 & -5 & 4 \\ -1 & -6 & 3 & -6\end{pmatrix}$$ using

1. **State the problem:** Find the inverse of matrix $$A=\begin{bmatrix}2 & 0 & 6 \\ 3 & 2 & 7 \\ 1 & 1 & 3\end{bmatrix}$$ using the adjoint method. 2. **Formula and rules:** The i

1. **State the problem:** Find the eigenvalues $\lambda_1$ and $\lambda_2$ of the matrix $$\begin{bmatrix} 3 & 3 \\ 3 & 4 \end{bmatrix}.$$\n\n2. **Formula:** Eigenvalues satisfy th

1. **Problem statement:** Given matrices $$A=\begin{pmatrix}1 & -2 & 3 \\ 4 & 0 & -2\end{pmatrix}, B=\begin{pmatrix}3 & 0 & 2 \\ 4 & 1 & 5\end{pmatrix}, C=\begin{pmatrix}1 & 3 \\ 2

1. Let's start by defining the problem: Given a square matrix $A$, we want to find its cofactor matrix, adjoint (also called adjugate) matrix, and inverse matrix. 2. The cofactor m

1. **State the problem:** Find the inverse of matrix $$A = \begin{bmatrix}4 & 4 & 2 & -1 \\ -1 & -4 & -4 & 3 \\ -4 & -4 & 3 & -1 \\ 2 & -2 & -3 & 2\end{bmatrix}$$ using the adjoint

1. **Problem statement:** Find the generalized eigenspaces $G(3,T)$ and $G(5,T)$, the characteristic and minimal polynomials of $T$, and Jordan bases corresponding to two given Jor

1. **Problem 1: Find rank, nullity, and kernel of the linear transformation** Given linear transformation $T: \mathbb{R}^6 \to \mathbb{R}^4$ defined by matrix

1. **Problem 1: Find rank, nullity, and kernel of the linear transformation** Given linear transformation $$T: \mathbb{R}^6 \to \mathbb{R}^4$$ defined by