Math Problem Statement

Prove that the set W = {(x, y, z) ∈ R³ : ax + by + cz = 0}, where a, b, c are scalars, is a subspace of R³.

Solution

To prove that WW is a subspace of R3\mathbb{R}^3, let’s proceed step by step:


Problem Restatement:

Let W={(x,y,z)R3:ax+by+cz=0}W = \{(x, y, z) \in \mathbb{R}^3 : ax + by + cz = 0\}, where a,b,ca, b, c are scalars. Prove that WW is a subspace of R3\mathbb{R}^3.


Subspace Conditions:

To verify that WW is a subspace of R3\mathbb{R}^3, we need to check the following three properties:

  1. The Zero Vector is in WW:

    • The zero vector in R3\mathbb{R}^3 is (0,0,0)(0, 0, 0). Substituting into the equation ax+by+cz=0ax + by + cz = 0: a(0)+b(0)+c(0)=0,a(0) + b(0) + c(0) = 0, which is true. Thus, the zero vector is in WW.
  2. Closure under Addition:

    • Let (x1,y1,z1)W(x_1, y_1, z_1) \in W and (x2,y2,z2)W(x_2, y_2, z_2) \in W. This implies: ax1+by1+cz1=0andax2+by2+cz2=0.a x_1 + b y_1 + c z_1 = 0 \quad \text{and} \quad a x_2 + b y_2 + c z_2 = 0. For their sum (x1+x2,y1+y2,z1+z2)(x_1 + x_2, y_1 + y_2, z_1 + z_2), we have: a(x1+x2)+b(y1+y2)+c(z1+z2)=(ax1+by1+cz1)+(ax2+by2+cz2)=0+0=0.a(x_1 + x_2) + b(y_1 + y_2) + c(z_1 + z_2) = (a x_1 + b y_1 + c z_1) + (a x_2 + b y_2 + c z_2) = 0 + 0 = 0. Hence, WW is closed under addition.
  3. Closure under Scalar Multiplication:

    • Let (x,y,z)W(x, y, z) \in W and kk be any scalar. Then: ax+by+cz=0.ax + by + cz = 0. For k(x,y,z)=(kx,ky,kz)k(x, y, z) = (kx, ky, kz), substituting into the defining equation of WW: a(kx)+b(ky)+c(kz)=k(ax+by+cz)=k(0)=0.a(kx) + b(ky) + c(kz) = k(ax + by + cz) = k(0) = 0. Thus, WW is closed under scalar multiplication.

Conclusion:

Since WW satisfies all three conditions, WW is a subspace of R3\mathbb{R}^3.


Would you like a further breakdown of any specific step or additional examples?
Here are some related questions to explore:

  1. What is the dimension of WW in R3\mathbb{R}^3 for specific values of a,b,ca, b, c?
  2. How does the condition ax+by+cz=0ax + by + cz = 0 define a geometric object in R3\mathbb{R}^3?
  3. How would the proof change if WW were a subset of Rn\mathbb{R}^n?
  4. Can WW ever represent a line in R3\mathbb{R}^3?
  5. What happens to WW if a=b=c=0a = b = c = 0?

Tip: Always confirm closure properties by substituting directly into the defining condition of the subspace.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Subspaces
Vector Spaces

Formulas

ax + by + cz = 0

Theorems

Subspace Test
Closure under addition and scalar multiplication

Suitable Grade Level

Undergraduate - Linear Algebra