📘 linear algebra

1. **Problem Statement:** Given vectors $v_1$ and $v_2$, list seven vectors in the span of $\{v_1, v_2\}$, show the weights used to generate each vector, and list their entries. Al

1. **State the problem:** Given two vectors $\mathbf{v}_1 = \begin{pmatrix}5 \\ -1 \\ 3\end{pmatrix}$ and $\mathbf{v}_2 = \begin{pmatrix}1 \\ 1 \\ -5\end{pmatrix}$, we want to anal

1. The problem is to understand the matrix \(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\). 2. This is a 2x2 matrix with elements \(a, b, c, d\) arranged in two rows and two columns

1. **Problem:** Find the inverse of the matrix $A$ using the adjoint method. 2. **Formula:** The inverse of a matrix $A$ is given by

1. **State the problem:** We need to show that the transformation $T : \mathbb{R}^3 \to \mathbb{R}^2$ defined by $T(x, y, z) = (z, x + y)$ is linear. 2. **Recall the definition of

1. The problem asks to define the span of a vector space. 2. The span of a set of vectors is the collection of all possible linear combinations of those vectors.

1. The problem is to define the span of a vector space. 2. The span of a set of vectors is the collection of all possible linear combinations of those vectors.

1. **State the problem:** Determine if the vectors $U_1 = (1, -2, 3, 4)$, $U_2 = (-2, 4, -1, -3)$, and $U_3 = (-1, 2, 7, 6)$ are linearly dependent or independent. 2. **Recall the

1. **State the problem:** Determine if the vector $\mathbf{v} = (2, 4, 6, 7, 8)$ is in the subspace of $\mathbb{R}^5$ spanned by the vectors $\mathbf{u}_1 = (1, 2, 0, 3, 0)$, $\mat

1. **State the problem:** Find the inverse of matrix $A = \begin{pmatrix}-2 & 1 & 3 \\ 5 & 0 & 2 \\ 1 & 6 & -3\end{pmatrix}$. 2. **Formula and rules:** The inverse of a matrix $A$,

1. **Problem Statement:** Find the inverse of matrix $A = \begin{pmatrix}-2 & 1 & 3 \\ 5 & 0 & 2 \\ 1 & 6 & -3\end{pmatrix}$. 2. **Formula and Rules:** The inverse of a matrix $A$,

1. **Problem Statement:** Determine if the vectors $\mathbf{u}_1 = (2, -1, 3, 2)$, $\mathbf{u}_2 = (1, 3, 4, 2)$, and $\mathbf{u}_3 = (3, -5, 2, 2)$ are linearly dependent or indep

1. **Problem Statement:** Solve the linear system using LU decomposition:

1. **State the problem:** Show that the set $W = \{(x,y,z) \in \mathbb{R}^3 : x + y + z = 0\}$ is a subspace of $\mathbb{R}^3$. 2. **Recall the subspace criteria:** A subset $W$ of

1. Problem statement: Multiply the matrices A and B and compute AB. A = $$\begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix}$$.

1. Let's solve a matrix multiplication example. 2. Problem: Multiply matrices $$A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$$ and $$B = \begin{bmatrix}5 & 6 \\ 7 & 8\end{bmatrix

1. Stating the problem. Subtract the following matrices and compute $A - B$.

1. Let's solve a simple example of matrix multiplication with two 2x2 matrices. 2. Suppose we have matrices:

1. **Matrix Addition and Scalar Multiplication** Given matrices:

1. **Statement of the problem:** Find the determinant of matrix $$A = \begin{bmatrix} 3 & 6 \\ 2 & 4 \end{bmatrix}$$.

1. **Problem statement:** Show that the set $S = \{p_1, p_2, p_3\}$ with $p_1 = (1, 2, 3)$, $p_2 = (-4, 5, 6)$, and $p_3 = (7, -8, 9)$ is a basis for $\mathbb{R}^3$.