Suppose ∪ and W are subspace af V. Such that dim∪=5,dim w=6 and dim v=9. Find dim (∪∩W).

Step1(1)&nbsp;∪&nbsp;and W are subspace af Vdim(∪+W)≤dim⁡(V)(2)&nbsp;dim∪+dim⁡W+dim⁡(∪+W)+dim⁡(∪∩W)(3)&nbsp;dim⁡(∪∩W)≤min{dim∪,dim⁡W}Step 2dim⁡cudim⁡W=dim⁡(∪+W)+dim⁡(∪∩W)Also,dim⁡(∪+W)≤dim⁡(V)dim⁡(∪+W)≤q⇒dim∪+dim⁡W−dim⁡(∪∩W)≤q⇒5+6−dim⁡(∪∩W)≤q⇒11−9≤dim⁡(∪∩W)⇒dim⁡(∪∩W)≤22≤dim⁡(∪∩W)≤min{dim∪,dim⁡W}2≤dim⁡(∪∩W)≤min{5,6}2≤dim(∪∩W)≤5

"Suppose ∪ and W are subspace af V...." solved and explained

High School
Algebra
Algebra I
Systems of equations

Petrolovujhm

Answered question

2022-04-03

Suppose

\cup

and W are subspace af V.
Such that

\dim \cup = 5

,dim

w = 6

and dim

v = 9

. Find dim (

\cup \cap

W).

Answer & Explanation

pypberissootcu

Beginner2022-04-04Added 14 answers

Step1
(1) $\cup$ and W are subspace af V
dim $(\cup + W) \leq \dim (V)$
(2) $\dim \cup + \dim W + \dim (\cup + W) + \dim (\cup \cap W)$
(3) $\dim (\cup \cap W) \leq min {\dim \cup, \dim W}$
Step 2
$\dim c u \dim W = \dim (\cup + W) + \dim (\cup \cap W)$
Also, $\dim (\cup + W) \leq \dim (V)$
$\dim (\cup + W) \leq q$
$\Rightarrow \dim \cup + \dim W - \dim (\cup \cap W) \leq q$
$\Rightarrow 5 + 6 - \dim (\cup \cap W) \leq q$
$\Rightarrow 11 - 9 \leq \dim (\cup \cap W)$
$\Rightarrow \dim (\cup \cap W) \leq 2$
$2 \leq \dim (\cup \cap W) \leq min {\dim \cup, \dim W}$
$2 \leq \dim (\cup \cap W) \leq min {5, 6}$
$2 \leq \dim (\cup \cap W) \leq 5$

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