Question

Theorem 1 Let $R, S \subseteq \mathbb{R}^n$. Then 1. $S^\perp \subseteq \mathbb{R}^n$. 2. $S \oplus S^\perp = \mathbb{R}^n$. 3. $(S^\perp)^\perp = S$. 4. $R \subseteq S$ if and only if $S^\perp \subseteq R^\perp$. 5. $(R+S)^\perp = R^\perp \cap S^\perp$. 6. $(R \cap S)^\perp = R^\perp + S^\perp$.

          Theorem 1
Let $R, S \subseteq \mathbb{R}^n$. Then
1. $S^\perp \subseteq \mathbb{R}^n$.
2. $S \oplus S^\perp = \mathbb{R}^n$.
3. $(S^\perp)^\perp = S$.
4. $R \subseteq S$ if and only if $S^\perp \subseteq R^\perp$.
5. $(R+S)^\perp = R^\perp \cap S^\perp$.
6. $(R \cap S)^\perp = R^\perp + S^\perp$.

Theorem 1
Let R, S ⊆ℝ^n. Then
1. S^⊥⊆ℝ^n.
2. S ⊕ S^⊥ = ℝ^n.
3. (S^⊥)^⊥ = S.
4. R ⊆ S if and only if S^⊥⊆ R^⊥.
5. (R+S)^⊥ = R^⊥∩ S^⊥.
6. (R ∩ S)^⊥ = R^⊥ + S^⊥.

Added by Samuel P.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sandip Ranjan

07/23/2021

Step 1

This is a fundamental property of vector spaces. Show more…

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Could you please prove the above questions one by one very clearly? Thank you, sir. Theorem 1: Let R, P ⊂ Rⁿ. Then 0 ∈ Rⁿ. 0 ≠ 0 = Rⁿ 0(g) = 9. Q ⊂ R if and only if S ⊂ R- 0(R+S) = RⁿS¹ 0(RⁿS) = R+S¹