Let V be the vector space of all real polynomials. Consider the subspace W spanned by t2 + t +2, t2 +2t +5, 5t2 +3t + 4, and 2t2 +2t+ 4. Then the dimension of W is,a)4b)3c)2d)1Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Explanation:

To find the dimension of the subspace W, we need to determine the number of linearly independent vectors in the span.

Step 1: Write the vectors in matrix form
The vectors can be written in matrix form as follows:

```
| 1 1 2 |
| 0 2 -1 |
| 0 5 4 |
```

Step 2: Perform row operations to reduce the matrix to row-echelon form
Performing row operations on the matrix, we can reduce it to row-echelon form:

```
| 1 1 2 |
| 0 2 -1 |
| 0 0 5 |
```

Step 3: Count the number of non-zero rows
The number of non-zero rows in the row-echelon form is equal to the number of linearly independent vectors. In this case, there are 2 non-zero rows.

Step 4: Conclusion
Therefore, the dimension of the subspace W is 2.

Answer: The correct answer is option C) 2.

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