¿Es subespacio Vectorial? Espacios de matrices, Álgebra Lineal
Introduction to the Exercise
- The video focuses on determining if set W is a vector subspace of 2x2 matrices with real entries.
- The matrices in W have the form beginpmatrix a & -b b & a endpmatrix , where a and b are specific numbers.
- Examples of matrices in W include combinations of positive and negative values for a and b .
Conditions for Subspace Verification
- To prove W is a subspace, show that the sum of two elements remains in W.
- Also, demonstrate that multiplying an element by any scalar keeps it within W.
- If either condition fails, or if zero vector isn't included, then W cannot be a subspace.
Summing Two Elements from Set W
- Select two general matrices from W: A_1 = beginpmatrix a_1 & -b_1 b_1 & a_1 endpmatrix , A_2 = beginpmatrix a_2 & -b_2 b_2 & a_2 endpmatrix .
- Their sum results in another matrix that meets the conditions for membership in set W.
- Both diagonal elements must be equal; off-diagonal elements must be negatives of each other.
Scalar Multiplication Check
- Take an element from W and multiply it by any scalar (denoted as α).
- Verify if the resulting matrix still satisfies the conditions to belong to set W.
- Confirmed that both conditions hold true; thus, conclude that W is indeed a subspace.
Exploring Inciso B Conditions
- In inciso B, matrices must satisfy a + d = 1 ; diagonal elements sum to one.
- Examples include matrices like beginpmatrix 1 & 5 -7 & 0 endpmatrix .
Understanding Subspaces and Zero Vector
- Clarification on zero vector: Having the zero vector in a set does not guarantee it's a subspace.
- If a set lacks the zero vector, it cannot be a subspace; presence of zero is mandatory for subspaces.
- Example provided: Matrix cer0 not belonging to W indicates W is not a subspace.
Properties of Subspaces
- New condition: Product of diagonal elements must equal zero for matrices in W.
- To confirm W as a subspace, check closure under addition and scalar multiplication.
- Example matrices are presented to test if their sum remains within W.
Counterexample for Closure
- Found two matrices in W whose sum does not belong to W, violating closure property.
- This counterexample demonstrates that W is not a subspace due to failure in closure under addition.
Exploring Further Conditions
- Inciso D prompts viewers to pause and attempt solving before revealing answers.
- The presence of the zero matrix alone does not imply that W is a subspace; further checks needed.
Testing with Examples
- Attempting various matrix combinations shows consistent results regarding membership in W.
- Finding pairs where one matrix's element negates another ensures their sum equals zero.
Conclusion on Subspace Verification
- Finding examples isn't sufficient; need general proof using variables for all elements in W.
Subspace Conditions
- The sum of elements a_1 + a_2 results in B_1 + b_2 and c_1 + C_2 , demonstrating closure under addition.
- Multiplying by a scalar alpha yields alpha a_1, alpha B_1, -alpha a_1 , confirming closure under scalar multiplication.