📘 Linear Algebra

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 two vectors $\mathbf{v}$ and $\mathbf{w}$ and need to compute their sum $\mathbf{v} + \mathbf{w}$ and their difference $\mathbf{v} - \mathbf{

1. **State the problem:** We need to multiply the matrix $$\left(\begin{array}{cc} \frac{-3}{7} & \frac{-1}{7} \\ \frac{-1}{7} & \frac{2}{7} \end{array}\right)$$

1. **Problem Statement:** Find the row echelon form of the matrix $$\begin{bmatrix} 1 & 2 & 0 & 1 & 2 \\ 3 & 6 & 2 & 5 & 0 \\ 0 & 2 & 1 & 5 & 2 \\ 2 & 3 & 7 & 1 & -1 \end{bmatrix}$

1. **State the problem:** We are given two matrices $$A = \begin{pmatrix} 6 & 8 \\ 10 & 6 \end{pmatrix}$$

1. **State the problem:** We are given the augmented matrix $$\begin{bmatrix} 2 & 3 & -1 & 9 \\ 1 & -1 & 2 & -3 \\ 3 & 1 & 3 & 2 \end{bmatrix}$$

1. The problem is to understand the expression $A*$. 2. In mathematics, the symbol $*$ often denotes an operation such as multiplication or a special operator depending on context.

1. The problem asks to find the adjoint (also called the conjugate transpose) of matrix $A$, denoted as $A^*$. 2. The adjoint $A^*$ of a matrix $A$ is found by taking the transpose

1. **State the problem:** Multiply the matrices $$A = \begin{pmatrix}1 & 5 \\ 3 & 3 \\ 0 & 0\end{pmatrix}$$

1. **State the problem:** Multiply the matrices $$A = \begin{bmatrix}1 & 5 \\ 3 & 3 \\ 0 & 0\end{bmatrix}$$

1. **Problem Statement:** Diagonalise the matrix $$A = \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix}$$ by an orthogonal transformation. 2. **Key Concept:**

1. **Find the inverse of matrix A:** Given matrix

1. **Problem statement:** (a) Given a linear transformation $f : \mathbb{R}^2 \to \mathbb{R}^3$ with

1. **Problem statement:** Given a rectangular matrix $$A = \begin{bmatrix} a & d \\ b & e \\ c & f \end{bmatrix},$$ compute the matrix product $$A^T A$$ and show that it is a symme

1. **Problem Statement:** Prove that for matrices $A$ and $B$, the transpose of their product satisfies $$(AB)^T = B^T A^T.$$ 2. **Definitions and Setup:** Let $A$ be an $m \times

1. **State the problem:** Compute the dot product of the vectors $\mathbf{a} = [14, 21, 28]$ and $\mathbf{b} = [4, 8, 20]$. 2. **Recall the formula:** The dot product of two vector

1. **State the problem:** Compute the norm (magnitude) of the vector $\mathbf{v} = [42, 14]$ using properties of the dot product and norm. 2. **Recall the formula:** The norm of a

1. **State the problem:** We are given vectors $\mathbf{u} = [-1, 3, -2]$, $\mathbf{v} = [4, 0, -1]$, and $\mathbf{w} = [-3, -1, 2]$. We need to compute the vector $3\mathbf{u} - 2

1. **Problem Statement:** We will solve several matrix operation problems including identity matrices, matrix multiplication, verification of matrix equality, transpose properties,

1. The problem is to find the determinant of a given matrix. 2. The determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain prop

1. **State the problem:** Find the matrix \( A = \begin{bmatrix} 1 & 9 & 3 \\ 2 & 5 & 4 \\ 3 & 7 & 8 \end{bmatrix} \). 2. **Understanding the matrix:** This is a 3x3 matrix with el