📘 linear algebra

1. **State the problem:** Find the transpose of the matrix $$A = \begin{bmatrix} 1 & 2 & 1 \\ -3 & -2 & 9 \\ -5 & 7 & -3 \end{bmatrix}$$

1. **Problem Statement:** We have data from 6 projects with developers $x_1$, complexity $x_2$, and time $y$. The model is $y_{pred} = b_0 + b_1 x_1 + b_2 x_2$.

1. The problem asks to write the vector $(0,0,0,0,0,0,0,0,0,10)$ in builder notation. 2. Builder notation for vectors typically expresses the vector as a linear combination of unit

1. **State the problem:** We need to find the coordinates of point $P$ given by the vector equation $$P = 5a - 2b$$ where $$a = \begin{pmatrix}3 \\ 2\end{pmatrix}$$ and $$b = \begi

1. **Problem statement:** Consider subspaces \(U = \{(x,0,y): x,y \in \mathbb{R}\}\) and \(W = \{(0,x,y): x,y \in \mathbb{R}\}\) of \(V = \mathbb{R}^3\). Show that \(\mathbb{R}^3 =

1. Diberikan matriks-matriks: $$A=\begin{bmatrix}6 & 1 \\ -3 & 2\end{bmatrix}, B=\begin{bmatrix}3 & 0 \\ 2 & 3\end{bmatrix}, C=\begin{bmatrix}2 & 3 & 5 \\ 1 & 6 & 2 \\ -1 & 1 & 0\e

1. **Problem a:** Write $v=(-5,4,9)$ as a linear combination of $u_1=(4,1,0)$ and $u_2=(1,2,3)$. 2. We want to find scalars $k_1$ and $k_2$ such that:

1. Problem statement: Given the matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ compute $F(-3)$ where $F(t)=\det(A - tI)$.\n2. Formula used: For a 2x2 matrix $A=\begin{bmatrix}a&b\\c

1. **Problem statement:** Given matrices $$X=\begin{pmatrix}1 & 2 \\ 3 & 0\end{pmatrix}, Y=\begin{pmatrix}4 & 1 \\ 2 & 5\end{pmatrix}, Z=\begin{pmatrix}0 & 3 \\ 1 & 2\end{pmatrix}$

1. Diketahui basis \(\vec{u}_1 = (1,1,0), \vec{u}_2 = (0,1,1), \vec{u}_3 = (1,0,1)\) di \(\mathbb{R}^3\) dengan hasil kali dalam Euclides. Tugas kita adalah menggunakan proses Gram

1. The problem asks to find an expression equivalent to the vector \( \left( \begin{array}{l} r \\ t \end{array} \right) \).\n\n2. This vector represents a point or direction in 2D

1. **Problem Statement:** Find $P(C^{-1})$ where $P(x) = 2x^2 - 4x - 2I$ and

1. The problem is to understand and work with matrices. 2. A matrix is a rectangular array of numbers arranged in rows and columns.

1. The problem describes a system of components and blends connected by variables $x_{ij}$, where $i$ indexes components and $j$ indexes blends. 2. To analyze or solve such a syste

1. The problem asks us to verify the truth of two matrix equalities for square matrices $M$ and $N$ of the same size. 2. For part (a), the expression is $M^2 - N^2 = (M + N)(M - N)

1. **State the problem:** Given matrices $$A = \begin{bmatrix} 1 & -1 \\ 0 & 2 \\ -3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$$

1. The problem asks for the condition on $a$ such that the matrix $$ A = \begin{bmatrix} -6 & a & a \\ 2 & -1 & 0 \\ -2 & 1 & 5 \end{bmatrix} $$

1. We are given the matrix $$A = \begin{bmatrix} 1 & 0 & -2 & -1 \\ 1 & 1 & 1 & 4 \\ 0 & 1 & 2 & 1 \\ -1 & 0 & 3 & 8 \end{bmatrix}$$

1. The problem asks why a transformed vector is a scalar multiple of the vector representing the twinned line. 2. When a vector is transformed by a linear transformation and the re

1. **Problem statement:** We have two linear transformations represented by matrices:

1. **State the problem:** We have a linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ defined by its action on the standard basis vectors: $$T(1,0,0) = (2,2,3), \quad T(0,1,