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Example 1: If w = $sp(v_1, v_2, \cdots, v_m)$ where $v_1, \cdots, v_m \in R^n$, then W is a subspace of $R^n$

sol:

(i) $v_i \in W$ therefore W is a nonempty subset of $R^n$

(ii) Let vectors u = $r_1v_1 + \cdots + r_nv_m \in W$ and v = $s_1v_1 + \cdots + s_nv_m \in W$ were $r,s \in R$.

$$u + v = (r_1+s_1)v_1 + \cdots + (r_n + s_n)v_m$$ which is in W therefore it is closed under vector addition.

(iii) Let $r \in R$, then rv = $rv_1 + \cdots + rv_m$ which is in W therefore W is a subspace of $R^n$

Example 2: If W = $\{ [x_1, x_2, x_3, x_4] \in R^4 \bigg| x_1 = x_3 - x_4, x_2 = x_3 + x_4\}$ Determine if W is a subspace of R^n

sol:

(i) Because [0,0,0,0] is in W therefore W is a nonempty subset of $R^4$

(ii) Let vectors u = $[a-b,a+b,a,b]$ and v = $[c-d, c+d, c,d], v,u \in W$

$$v + u = [(a+c) - (b+d), (a+c) + (b+d), (a+c), (b+d)]$$ This is in the form of W therefore its closed under vector addition

(iii) Let $r \in R$, then ru = $[ra-rb, ra+rb, ra,rb]$ Therefore it is also closed under scalar multiplication because of the form therefore W is a subspace of $R^4$

Example 3: Is $W = \{[x_1,x_2,x_3] \in R^3 \bigg| x_1 + x_3 = x_2 + 3\}$ a subspace of $R^3$.

Sol:

No because [0,0,0] is not in W. Counter example:

u = $[x_1, x_2, x_3]$ , $u \in W$

$0u = [0,-3,0]$

1 Answers 1

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This is correct. A little bit of nitpicking: in example 1, when you show that $rv \in W$ for $r \in R, v \in W$, you should use a general $v$ (as you did tot show that $u+ v \in W$, but the fact that you did it there makes me think you know this). And for example 3: you are done after your remark that the zerovector is not in $W$. Note that $0u = [0,0,0]\neq [0,-3,0]$ as you wrote!