📘 linear algebra

1. **Problem:** Determine if the set of all vectors of the form $(a,0,0)$ in $\mathbb{R}^3$ is a subspace. 2. **Subspace Test:** A subset $W$ of $\mathbb{R}^3$ is a subspace if it

1. The statement says: If $\{ \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \}$ is a basis for a vector space $V$, then for every vector $\mathbf{v}$ in $V$ there exists a uniqu

1. **Problem:** Determine whether each statement about basis and dimension is True (T) or False (F). 2. **Recall important definitions and facts:**

1. **State the problem:** Find the least squares solution $\hat{x}$ to the system $A x = b$ where $$

1. **State the problem:** Find the least squares solution $\hat{x}$ to the system $Ax = b$ where $$A = \begin{pmatrix} 1 & 3 & 3 \\ 5 & 4 & 2 \\ 2 & 2 & 2 \\ 4 & 2 & 5 \\ 6 & 5 & 5

1. **State the problem:** Find the least squares solution $\hat{x}$ to the system $A x = b$ where $$A = \begin{pmatrix} 1 & 3 & 3 \\ 5 & 4 & 2 \\ 2 & 2 & 2 \\ 4 & 2 & 5 \\ 6 & 5 &

1. **State the problem:** Find the least squares solution $\hat{x}$ to the system $A x = b$ where $$A = \begin{pmatrix} 1 & 3 & 3 \\ 5 & 4 & 2 \\ 2 & 2 & 2 \\ 4 & 2 & 5 \\ 6 & 5 &

1. **State the problem:** We have two matrices: $$\begin{bmatrix} 1 & -3 & 3 & 0 \\ 5 & -2 & 2 & -5 \\ 0 & 4 & -2 & 5 \end{bmatrix} \quad \to \quad \begin{bmatrix} 1 & -3 & 3 & 0 \

1. **Problem Statement:** Find the row space, column space, and null space of the matrix

1. **State the problem:** We are given vectors $v_1, v_2, v_3, v_4, v_5$ in $\mathbb{R}^4$ and need to find:

1. The problem involves multiplying a 5x1 column vector by a 1x2 row vector. 2. The column vector is $$\begin{bmatrix}0 \\ 0 \\ 0 \\ 3 \\ 2\end{bmatrix}$$ and the row vector is $$\

1. **Problem Statement:** Find the rank of the matrix $$\begin{bmatrix} 6 & 1 & 3 & 8 \\ 4 & 2 & 6 & -1 \\ 10 & 3 & 9 & 7 \\ 16 & 4 & 12 & 15 \end{bmatrix}$$

1. **Determine whether each set is a subspace of $P_2$** (the space of all polynomials of degree at most 2). Recall: A subset $W$ of a vector space $V$ is a subspace if it satisfie

1. **State the problem:** Find the value of the determinant of the matrix $$\begin{pmatrix}3 & 1 & 2 \\ 1 & 4 & 2 \end{pmatrix}$$

1. **State the problem:** Find the least squares solution $\hat{x}$ to the system $A x = b$ where $$A = \begin{pmatrix}1 & 5 & 3 \\ 3 & 5 & 3 \\ 5 & 2 & 4 \\ 5 & 5 & 4 \\ 1 & 3 & 3

1. **State the problem:** Find the determinant of the given 2x2 matrix $$\begin{bmatrix}-9 & 6 \\ 4 & -8\end{bmatrix}$$

1. The problem is to determine if the vector $\mathbf{u}$ is a linear combination of the vectors $\mathbf{v_1}$, $\mathbf{v_2}$, and $\mathbf{v_3}$. 2. A vector $\mathbf{u}$ is a l

1. The problem is to understand why no two different matrices can be equal. 2. By definition, two matrices are equal if and only if they have the same dimensions and all correspond

1. **Problem Statement:** How to multiply two matrices. 2. **Matrix Multiplication Rule:** To multiply two matrices $A$ and $B$, the number of columns in $A$ must equal the number

1. The problem asks to find $A^2$ for the matrix $A = \begin{pmatrix} x & 0 \\ 5 & y \end{pmatrix}$. 2. To find $A^2$, we multiply matrix $A$ by itself: $$A^2 = A \times A = \begin

1. **Problem Statement:** We are given a system of simultaneous linear equations representing reservoir simulation: $$a_i P_{i-1} + b_i P_i + C_i P_{i+1} = d_i, \quad i=1,2,\ldots,