📘 linear algebra

1. **Problem:** Find the eigenvalues of matrix $$A = \begin{bmatrix}1 & 2 \\ 5 & 4\end{bmatrix}$$. 2. **Formula:** Eigenvalues $$\lambda$$ satisfy $$\det(A - \lambda I) = 0$$ where

1. **State the problem:** We are given an inner product on $\mathbb{R}^4$ defined by $\langle x,y \rangle = x^T M y$ where $$M = \begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 3 & 0 & 0 \\

1. **Problem statement:** We are given an inner product on $\mathbb{R}^4$ defined by $$\langle x,y \rangle = x^T M y,$$ where $$M = \begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 3 & 0 & 0

1. **Problem Statement:** Verify that the function $\langle x,y \rangle = x^T M y$ defines an inner product on $\mathbb{R}^4$ where $$M = \begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 3 &

1. **Problem statement:** We are given the system of linear equations: $$\begin{cases} 4x + y + z = 6 \\ x + 3y + z = 5 \\ x + y + 2z = 4 \end{cases}$$

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

1. **State the problem:** We have the system of linear equations: $$\begin{cases} 4x + y + z = 6 \\ x + 3y + z = 5 \\ x + y + 2z = 4 \end{cases}$$

1. **State the problem:** We need to find the eigen-decomposition of the coefficient matrix $A$ of a system of linear equations and then use this decomposition to solve the system.

1. **State the problem:** Solve the system of linear equations given by the matrix equation $$\begin{bmatrix}3 & -1 & 0 \\ -1 & 3 & -1 \\ 0 & -1 & 3\end{bmatrix} \begin{bmatrix}p \

1. **Problem Statement:** Determine which pairs of vectors from the given list are orthogonal. Two vectors are orthogonal if their dot product equals zero. 2. **Formula:** The dot

1. **Problem:** Show that the matrix $$A=\begin{pmatrix} \alpha + i\gamma & -\beta + i\delta \\ \beta + i\delta & \alpha - i\gamma \end{pmatrix}$$ is unitary if $$\alpha^2 + \beta^

1. **Problem Statement:** Find the rank of matrix $A = \begin{bmatrix}1 & 2 & 3 \\ 3 & 4 & 5 \\ 3 & 2 & 1\end{bmatrix}$. 2. **Recall:** The rank of a matrix is the maximum number o

1. **Problem Statement:** Multiply the two matrices $$A = \begin{pmatrix} 1 & 0 & 3 \\ 2 & 1 & 2 \\ 1 & 3 & 1 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 2 & 2 & 0 \\

1. Problem: Suppose that $T : \mathbb{R}^5 \to \mathbb{R}^2$ and $T(x) = Ax$ for some matrix $A$ and each $x$ in $\mathbb{R}^5$. How many rows and columns does $A$ have? Step 1: Un

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

1. **Problem statement:** Given vectors $V_1 = (3, 2, 0)$, $V_2 = (1, 3, 0)$, $V_3 = (2, 5, 0)$, and $V_4 = (6, 1, 0)$ in the set $V = \{(a,b,0) \mid a,b \in \mathbb{R}\}$, find th

1. **Problem statement:** Solve the system of linear equations using row operations (Gaussian elimination). 2. **General approach:** We write the system as an augmented matrix and

1. **Problem statement:** Express the vectors as linear combinations of \(u = (2, 1, 4)\), \(v = (1, -1, 3)\), and \(w = (3, 2, 5)\). We want to find scalars \(a, b, c\) such that:

1. **Problem Statement:** Find the matrix $A$ representing the linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^3$ defined by $$T\begin{pmatrix}x \\ y\end{pmatrix} = \begin{pm

1. **Problem Statement:** Explain the definition of a diagonalizable operator and provide an example. 2. **Definition:** A linear operator $T: V \to V$ on a vector space $V$ is cal

1. **Problem statement:** Show that $\mathbb{R}^2$ is a normed linear space with the norm defined as $\|x\| = \max(|x_1|, |x_2|)$ for $x = (x_1, x_2) \in \mathbb{R}^2$. 2. **Defini