# Section 4.3 $\quad$ Subspaces (cont)

## Definition of Linear Combination

Let $\mathbf{v}_1$, $\mathbf{v}_2$, $\cdots$, $\mathbf{v}_k$ be vectors in a vector space $V$. <br /><br /><br /><br />

### Example 1

Every polynomial of degree $\leq 2$ is a linear combination of $t^2$, $t$, $1$.

### Example 2

Show that the set of all vectors in $\mathbb{R}^3$ of the form $\left[\begin{array}{c}a \\ b \\ a+b \end{array}\right]$ is a linear combination of $\mathbf{v}_1 = \left[\begin{array}{c}1 \\ 0 \\ 1 \end{array}\right]$ and $\mathbf{v}_2 = \left[\begin{array}{c}0 \\ 1 \\ 1 \end{array}\right]$.

### Example 3

In $\mathbb{R}^3$, let
\begin{equation*}
  \mathbf{v}_1 = \left[\begin{array}{c}1 \\ 2 \\ 1 \end{array}\right],~~~
  \mathbf{v}_2 = \left[\begin{array}{c}1 \\ 0 \\ 2 \end{array}\right],~~~
  \mathbf{v}_3 = \left[\begin{array}{c}1 \\ 1 \\ 0 \end{array}\right]
\end{equation*}
Verify that the vector
\begin{equation*}
\mathbf{v} = \left[\begin{array}{c}2 \\ 1 \\ 5 \end{array}\right]
\end{equation*}
is a linear combination of $\mathbf{v}_1$, $\mathbf{v}_2$, and $\mathbf{v}_3$.


```python
from sympy import *

a, b, c = symbols('a b c');

Eq1 = a + b + c - 2;
Eq2 = 2*a + c - 1;
Eq3 = a + 2*b - 5;

solve([Eq1, Eq2, Eq3], (a, b, c))
```




    {a: 1, b: 2, c: -1}



### Example 4

Consider the homogeneous system
$$A\mathbf{x} = \mathbf{0}$$
where $A$ is an $m\times n$ matrix. The set $W$ of solutions is a subset of $\mathbb{R}^n$. Verify that $W$ is a subspace of $\mathbb{R}^n$ (called **solution space**).

**Remark** The set of all solutions of the linear system $A\mathbf{x} = \mathbf{b}$, with $\mathbf{b} \neq \mathbf{0}$, is <br /><br /><br /><br />

77c2398c3b038694e26635a70f0890b5192ef6cc
ipynb
Jupyter Notebook
Jupyter_Notes/Lecture15_Sec4-3_SubspacesPart2.ipynb
xiuquan0418/MAT341
2fb7ec4e5f0771f10719cb5e4a00a7ab07c49b59
MIT
Jupyter_Notes/Lecture15_Sec4-3_SubspacesPart2.ipynb
xiuquan0418/MAT341
2fb7ec4e5f0771f10719cb5e4a00a7ab07c49b59
MIT
Jupyter_Notes/Lecture15_Sec4-3_SubspacesPart2.ipynb
xiuquan0418/MAT341
2fb7ec4e5f0771f10719cb5e4a00a7ab07c49b59
MIT
Qwen/Qwen-72B
1. YES
2. YES