📘 Linear Algebra

1. **Problem statement:** (a) Given the transformation matrix from frame A to frame B:

1. **State the problem:** Given matrix $A = \begin{bmatrix}5 & 0 \\ -6 & 2\end{bmatrix}$, find $A^2$ (the square of matrix $A$). 2. **Formula used:** To find $A^2$, we multiply mat

1. **State the problem:** We are given matrix $$A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$ with $$\det(A) = 4$$.

1. **Problem Statement:** We are given matrices A and B from problems (4) and (5) and asked to:

1. **State the problem:** Find the eigenvalues of the matrix $$A = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 6 & 0 \\ 8 & 0 & 5 \end{bmatrix}$$ given the expression $$3 + 8A - 5I$$ where $$

1. **State the problem:** We are given a matrix $$A = \begin{bmatrix}4 & 0 & 0 \\ 0 & 6 & 0 \\ 8 & 0 & 5\end{bmatrix}$$ and we want to analyze it, for example, find its determinant

1. **Problem Statement:** Find the eigenvalues of the matrix expression $$3 + 8A - 5I$$ where $$A = \begin{bmatrix}4 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\end{bmatrix}$$ and $$I$$ is the

1. **Problem statement:** Determine whether the vector $w = \begin{pmatrix} -2 \\ 1 \\ 2 \end{pmatrix}$ is in the range of the linear operator $T : \mathbb{R}^3 \to \mathbb{R}^3$ d

1. **Problem Statement:** Find the modal matrix and diagonalize the matrix $$A = \begin{bmatrix} 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \end{bmatrix}$$.

1. **Problem statement:** We have two planes in $\mathbb{R}^3$:

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:** Solve a system of linear equations using Cramer's Law. 2. **Formula:** For a system of $n$ linear equations with $n$ variables, Cramer's Law states that e

1. **Problem Statement:** We are given a matrix $$ A = \begin{bmatrix} 2 & -6 & -2 & -3 \\ 5 & -13 & -4 & -7 \\ -1 & 4 & 1 & 2 \\ 0 & 1 & 0 & 1 \end{bmatrix} $$

1. **State the problem:** Solve the system of linear equations using Gauss-Jordan elimination: $$\begin{cases} x_1 + 2x_2 - x_3 + x_4 = 1 \\ 3x_2 + 6x_3 - 3x_4 = -3 \end{cases}$$

1. **Problem:** Identify the matrix given: \[

1. **Problem statement:** Show that the reflection transformation $T:\mathbb{R}^2 \to \mathbb{R}^2$ about a line $\ell$ through the origin with slope $m \neq 0, \infty$ is linear,

1. The problem is to express a vector or a linear transformation in canonical form, which typically means representing it in a standard or simplest form. 2. In linear algebra, the

1. **Πρόβλημα:** Λύστε το σύστημα γραμμικών εξισώσεων με $A=3$ και $B=2$: $$\begin{cases} x_1 + 3x_3 = 3 \\ 2x_1 - x_2 + x_3 = 2 \\ -x_1 + 2x_2 - x_3 = 3 \end{cases}$$

1. **Problem statement:** Find the matrix \(\hat{P}\) representing the projection onto the line \(x + \sqrt{3} y = 0\) by considering the transformations of the unit vectors \(\hat

1. **Stating the problem:** We are asked to find the characteristic equations of given 2x2 matrices.

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