📘 linear algebra

1. Let's start by stating the problem: You want to understand the concept of the inner product in vector spaces. 2. An inner product is a function that takes two vectors from a vec

1. The problem involves vectors \(\mathbf{x}\) and \(\mathbf{y}\), which means we are dealing with vector operations rather than scalar algebra. 2. Common vector operations include

1. Let's start by understanding the problem: You want to know what the vector $\mathbf{x} + \mathbf{y}$ represents and how to find its length using the dot product. 2. The vector $

1. The problem is to understand why the norm is not linear on $\mathbb{R}^n$ and how the dot product helps introduce linearity. 2. The norm $\|\cdot\|$ on $\mathbb{R}^n$ is defined

1. **Problem Statement:** Find the eigenvalues and eigenvectors of the matrix $$A = \begin{bmatrix} -9 & 4 & 4 \\ -8 & 3 & 4 \\ -16 & 8 & 7 \end{bmatrix}$$

1. **Problem:** Solve the system $$\begin{cases} x_1 + x_2 = 2 \\ 5x_1 + 6x_2 = 9 \end{cases}$$

1. **Problem:** Solve the system \(x_1 + x_2 = 2\) and \(5x_1 + 6x_2 = 9\) by inverting the coefficient matrix. 2. **Formula:** For a system \(AX = B\), the solution is \(X = A^{-1

1. (i) **Problem:** Find $a$, $b$, $x$, and $y$ given the matrix equation $$\left(\begin{array}{cc} 3x + 4y & 6 \\ a + b & 2a - b \end{array}\right) \left(\begin{array}{c} x - 2y \

1. **Problem Statement:** Given the matrix $$A = \begin{bmatrix} 3 & 1 \\ 2 & 1 \end{bmatrix},$$ we need to compute the following:

1. **State the problem:** We are given the equation $$15 \cdot (7A)^{-1} = \begin{bmatrix}-3 & 7 \\ 1 & -2\end{bmatrix}$$ and need to find the matrix $$A$$. 2. **Recall the formula

1. We are given the inverse of matrix $A$, denoted as $A^{-1} = \begin{bmatrix} 2 & -1 \\ 3 & 5 \end{bmatrix}$, and we need to find matrix $A$. 2. Recall the property that $A \cdot

1. **State the problem:** We are given the inverse of matrix $A$, denoted as $A^{-1} = \begin{bmatrix} 2 & -1 \\ 3 & 5 \end{bmatrix}$, and we need to find the original matrix $A$.

1. **Problem Statement:** We are given two matrix inverse expressions:

1. **Problem Statement:** Verify that $$(ABC)^{-1} = C^{-1} B^{-1} A^{-1}$$ for the given matrices: $$A = \begin{bmatrix}3 & 1 \\ 5 & 2\end{bmatrix}, B = \begin{bmatrix}2 & -3 \\ 4

1. The problem is to find the equation or expression involving matrices A, B, and C given their values. 2. Typically, matrix problems involve operations like addition, subtraction,

1. **Problem Statement:** Verify the matrix identity $$(ABC)^{-1} = C^{-1} B^{-1} A^{-1}$$ where $A$, $B$, and $C$ are invertible matrices. 2. **Recall the formula for the inverse

1. **Problem Statement:** Find the inverse of the given matrices: Matrix 1:

1. Problem: Calculate total raw materials required and total cost for products A and B. Given matrix of materials per unit:

1. **Problem a:** Express $\vec{w} = (9, 2, 7)$ as a linear combination of $\vec{u} = (1, 2, -1)$ and $\vec{v} = (6, 4, 2)$. We want scalars $a$ and $b$ such that:

1. **State the problem:** Find the inverse of the given 3x3 matrices \(v\) and \(vi\).\n\n2. **Recall the formula:** The inverse of a matrix \(A\) is given by \(A^{-1} = \frac{1}{\

1. **Problem Statement:** Find the inverse of the given 3x3 matrices \(v\) and \(vi\).\n\n2. **Formula and Rules:** The inverse of a matrix \(A\) (if it exists) is given by \(A^{-1