Question 1174473: Consider the following subsets of the vector space R4?
S1={(1,0,1,1),(1,−1,1,1),(0,1,0,0),(1,0,1,0)}
S2={(1,0,1,2),(1,0,0,0),(0,0,2,−1)}
Which of the following statements is true?
Select one:
S2 is linearly independent
S1 is linearly independent
S1 spans R4
S2 spans R4
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's analyze each set to determine the correct answer.
**S1 = {(1, 0, 1, 1), (1, -1, 1, 1), (0, 1, 0, 0), (1, 0, 1, 0)}**
* **Linear Independence:**
* To check for linear independence, we can form a matrix with these vectors as rows (or columns) and calculate its determinant. If the determinant is non-zero, the vectors are linearly independent.
* Alternatively, we can check the rank of the matrix. If the rank of the matrix is equal to the number of vectors, then the vectors are linearly independent.
* The code provided shows that S1 is linearly dependent.
* **Spanning R4:**
* For a set of vectors to span R4, it must have at least 4 linearly independent vectors.
* Since S1 is linearly dependent, it cannot span R4.
**S2 = {(1, 0, 1, 2), (1, 0, 0, 0), (0, 0, 2, -1)}**
* **Linear Independence:**
* To check for linear independence, we can form a matrix with these vectors as rows (or columns) and check its rank.
* The code provided shows that S2 is linearly independent.
* **Spanning R4:**
* For a set of vectors to span R4, it must have at least 4 vectors.
* Since S2 has only 3 vectors, it cannot span R4.
**Conclusion**
* S1 is linearly dependent and does not span R4.
* S2 is linearly independent and does not span R4.
Therefore, the correct statement is:
* **S2 is linearly independent**
|
|
|
