Notes on Subspaces in R4

True-False Question

• Statement: The set V is a subspace of R4 (true or false).

Definition of Subset

• Subset: Any collection of vectors.

Definition of Subspace

• Subspace: A special subset of vectors that satisfies three conditions:
  1. Contains Zero Vector: Must include the zero vector.
  2. Closed Under Addition: If two vectors in the set are added, their sum must also be in the set.
  3. Closed Under Scalar Multiplication: If a vector in the set is multiplied by a scalar, the resulting vector must still be in the set.

Analyzing Set V

• Definition: Set V is defined by the condition:

  [ x - 4z = 0 ]

• This means that for a vector in V, the first entry minus four times the third entry equals zero.

Checking Conditions for Set V

1. Zero Vector:
   • Check if zero vector (0, 0, 0, 0) is in V.
   • Calculation:
     [ 0 - 4(0) = 0 ]
   • Conclusion: Yes, includes zero vector.
2. Closed Under Addition:
   • Define vectors A and B in V:
     • ( A: [A_1, A_2, A_3, A_4] )
     • ( B: [B_1, B_2, B_3, B_4] )
   • Conditions for A and B:
     • ( A_1 - 4A_3 = 0 )
     • ( B_1 - 4B_3 = 0 )
   • Check sum ( A + B ):
     • First entry: ( A_1 + B_1 )
     • Third entry: ( A_3 + B_3 )
   • Verify:
     • ( (A_1 + B_1) - 4(A_3 + B_3) = 0 )
   • Rearranging gives:
     • ( (A_1 - 4A_3) + (B_1 - 4B_3) = 0 )
   • Conclusion: Yes, it is closed under addition.
3. Closed Under Scalar Multiplication:
   • Define vector A in V: ( A: [A_1, A_2, A_3, A_4] )
   • Multiply by scalar C: ( C imes A = [CA_1, CA_2, CA_3, CA_4] )
   • Check if in V:
     • Verify: ( CA_1 - 4(CA_3) = 0 )
   • Factor out C:
     • ( C(A_1 - 4A_3) = 0 )
   • Since ( A_1 - 4A_3 = 0 ), it holds true for any C.
   • Conclusion: Yes, it is closed under scalar multiplication.

Final Conclusion

• Since set V satisfies all three conditions, it is confirmed that V is a subspace of R4.