📘 linear algebra

1. **Problem Statement:** We are given vectors and asked to find their norms (lengths), check if they are unit vectors, and normalize a vector.

1. **Problem Statement:** Given two vectors $u$ and $v$, find the value of $k$ such that their dot product $u \cdot v = 0$, meaning $u$ and $v$ are orthogonal. 2. **Vectors:**

1. **Stating the problem:** Given vectors \( u = (2,4,-5) \) and \( v = (1,-6,9) \), perform the following vector operations:

1. **Problem Statement:** Given the vector equation $(x - y, x + y, z - 1) = (4, 2, 3)$, find the values of $x$, $y$, and $z$.

1. **Problem Statement:** We have a network of one-way streets with unknown flow rates $x_1, x_2, x_3, x_4, x_5, x_6, x_7$ representing vehicles per hour. Given known flows and dir

1. **Problem:** Find the geometric and algebraic multiplicity of each eigenvalue of matrix \(A\), determine if \(A\) is diagonalizable, and if so, find matrix \(P\) that diagonaliz

1. **Vector Space:** A set of vectors where you can add any two vectors and multiply vectors by scalars, and the results stay in the set. 2. **Subspace:** A smaller vector space in

1. The problem asks to find the inverse of the matrix $$\begin{bmatrix}1 & 2 \\ 2 & 4\end{bmatrix}$$. 2. The formula for the inverse of a 2x2 matrix $$A = \begin{bmatrix}a & b \\ c

1. **Problem Statement:** 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 Impor

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. **Define the following with examples:** 1. Vector space: A set of vectors closed under addition and scalar multiplication. Example: $\mathbb{R}^2$ with usual addition and scalar

1. **Problem:** Find the order of $(AB)^t$ given $A = [a_{ij}]_{m \times n}$ and $B = [b_{ij}]_{n \times r}$. 2. **Recall:** The order (dimensions) of matrix $A$ is $m \times n$ an

1. The problem asks for the necessary conditions for the consistency of a non-homogeneous system of linear equations. 2. A non-homogeneous system can be written as $A\mathbf{x} = \

1. **Problem Statement:** Solve the systems of linear equations represented by the augmented matrices given in parts (a) and (b). 2. **Matrix (a):**

1. **State the problem:** We are given matrices and asked to evaluate the expression:

1. The problem asks to identify the condition under which a square matrix $A$ is singular. 2. A matrix is singular if it does not have an inverse.

1. The problem asks for the order of the product of two matrices $A$ and $B$ where $A$ is of order $m \times n$ and $B$ is of order $n \times p$. 2. The rule for matrix multiplicat

1. **Stating the problem:** We are given a system of inequalities:

1. **Problem statement:** Given matrix $$A=\begin{bmatrix}1 & 0 & 3 \\ 2 & 5 & 3 \\ 2 & 0 & 5 \end{bmatrix}$$, find the element at the intersection of row 1 and column 2 of the inv

1. Bài toán yêu cầu tìm hạng của ma trận $A$ kích thước $4 \times 4$ với điều kiện $\det(A) \neq 0$. 2. Công thức và kiến thức quan trọng:

1. Bài toán yêu cầu tìm hạng của ma trận $$A=\begin{bmatrix}1 & 5 & 3 & 7 \\ 2 & 9 & 6 & 8 \\ 0 & 0 & 12 & 9 \\ 0 & 0 & 4 & 6 \end{bmatrix}$$. 2. Hạng của ma trận là số lượng hàng