📘 linear algebra

1. **State the problem:** Compute the linear combination $u - 2v + 4w$ where $u = [1, 2, 1, 0]$,

1. **Stating the problem:** Cramer's Rule is a method to solve a system of linear equations using determinants. 2. **Formula and explanation:** For a system of $n$ equations with $

1. **State the problem:** We are given two vectors $\mathbf{v} = [2, -1]$ and $\mathbf{w} = [-3, -2]$. We need to compute the vector sums $\mathbf{v} + \mathbf{w}$ and $\mathbf{v}

1. **State the problem:** We are given vectors $v = [1, 3]$ and $w = [-2, 5]$ in $\mathbb{R}^2$. We want to find scalars $r$ and $s$ such that $$r v + s w = [-1, 19].$$ 2. **Set up

1. **Problem Statement:** Given vectors and matrices:

1. **State the problem:** We need to find the value of $k$ such that the vectors $\mathbf{v}$ and $\mathbf{u}$ are orthogonal. 2. **Recall the formula:** Two vectors $\mathbf{v} =

1. **State the problem:** Given velocity vectors $\mathbf{V}_A = 400\mathbf{i} + 100\mathbf{j}$ km/hr (where $\mathbf{i}$ points east and $\mathbf{j}$ points north), $\mathbf{V}_W

1. **State the problem:** We need to find all values of $k$ such that the product

1. **Problem Statement:** Find the specified rows and columns of the matrix products $AB$, $BA$, and $AA$ where

1. **Stating the problem:** We are given matrices A, B, C, D, and E and asked to compute various matrix operations including sums, differences, scalar multiples, transposes, and tr

1. a. Determine if $BA$ is defined and find its size. - Matrix $B$ is $4 \times 5$ and $A$ is $4 \times 5$.

1. **Problem:** Determine if the matrix product $BA$ is defined and find its size if it is. 2. **Given:**

1. **State the problem:** Calculate the vector expression $$3\mathbf{a} - \frac{1}{4}(\mathbf{c} - \mathbf{b})$$ where $$\mathbf{a} = \begin{pmatrix}-8 \\ 7\end{pmatrix}, \mathbf{b

1. The problem asks to write the augmented matrix for the system of linear equations: $$\begin{cases} 9x_1 + 6x_2 + 4x_3 = 8 \\ 7x_1 + x_2 - 9x_3 = 7 \end{cases}$$

1. **State the problem:** We are given an augmented matrix representing a system of linear equations (SLE) with parameter $k$. We want to find the value of $k$ such that the system

1. **Stating the problem:** We are given two theorems about vector spaces and bases:

1. Let's start by stating the problem: Find the eigenvalues of a given square matrix $A$. 2. The eigenvalues $\lambda$ of a matrix $A$ satisfy the characteristic equation:

1. **Problem statement:** Find vectors in $\mathbb{R}^4$ that form an orthogonal basis with the given vectors $\mathbf{v}_1 = (1, -2, 2, -3)$ and $\mathbf{v}_2 = (2, -3, 2, 4)$. Th

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 $