Linear Algebra: Spanning Set Theorem

Introduction to Spanning Set Theorem

• Spanning Set Theorem:
  • If a vector ( v_k ) in set ( S = {v_1, v_2, \ldots, v_p} ) is a linear combination of the remaining vectors, removing ( v_k ) still spans the same space ( H ).
  • If ( H ) is not just the zero vector, there exists a subset of ( H ) that forms a basis.

Proof of Spanning Set Theorem (Part 1)

• Assumption: ( S = {v_1, v_2, \ldots, v_p} ) and vector ( x ) is a linear combination of these vectors.
• Choose ( v_k ) as the last vector ( v_p ):
  • ( v_p ) is a linear combination of ( v_1, v_2, \ldots, v_{p-1} ).
• Substitute: Replace ( v_p ) in the expression for ( x ) to show that ( x ) can be written using ( {v_1, v_2, \ldots, v_{p-1}} ).
• Conclusion: The span remains the same as ( x ) is arbitrary, proving the spanning set reduction.

Basis from Spanning Set (Part 2)

• Conditions:
  1. ( S ) is linearly independent: ( S ) is a basis.
  2. ( S ) is linearly dependent: Rewrite a vector as a combination of others and remove it.
• Process:
  • If removing ( v_k ) results in a linearly independent set, a basis is found.
  • If not, repeat until achieving linear independence.
• Result: A basis is found by removing redundant vectors without reaching the zero vector.

Example: Finding Distinct Bases

• Vectors: ( v_1, v_2, v_3 ) with ( H ) being the span.
• Given: ( 4v_1 + 5v_2 - 3v_3 = \text{zero vector} ).
• Three Bases:
  1. Remove ( v_3 ): {v_1, v_2}.
  2. Remove ( v_2 ): {v_1, v_3}.
  3. Remove ( v_1 ): {v_2, v_3}.
• Note: All three are linearly independent and span ( H ).

Example Problem: Are ( v_1, v_2, v_3 ) a Basis for ( H )?

• Vectors: ( v_1 = [1, 0, 1], v_2 = [0, 1, 1], v_3 = [0, 1, 0] ).
• Space ( H ): Set of vectors ( [a, b, b] ).
• Linear Combination: Express ([a, b, b]) using ( v_1, v_2, v_3 ).
• Issue: ( v_3 = [0, 1, 0] ) is not in ( H ) because second and third elements are not equal.
• Conclusion: ( v_1, v_2, v_3 ) cannot be a basis for ( H ) since one vector ( (v_3) ) is not in ( H ).

End of Lecture: Questions can be asked in the comments for further clarification.