Question:
Consider the real vector space \( P_{2020} = \left\{ \sum_{i=0}^n a_i x^i : a_i \in \mathbb{R}, \, 0 \le n \le 2020 \right\}. \) Let \( W \) be the subspace given by
\[
W = \left\{ \sum_{i=0}^n a_i x^i \in P_{2020} : a_i = 0 \text{ for all odd } i \right\}.
\]
Then the dimension of \( W \) is ................
Consider the real vector space \( P_{2020} = \left\{ \sum_{i=0}^n a_i x^i : a_i \in \mathbb{R}, \, 0 \le n \le 2020 \right\}. \) Let \( W \) be the subspace given by
\[
W = \left\{ \sum_{i=0}^n a_i x^i \in P_{2020} : a_i = 0 \text{ for all odd } i \right\}.
\]
Then the dimension of \( W \) is ................
Show Hint
The dimension of a subspace equals the number of linearly independent basis vectors that satisfy the defining conditions.
Updated On: Dec 3, 2025
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Verified By Collegedunia
Correct Answer: 1011
Solution and Explanation
Step 1: Identify allowed coefficients.
Only even powers are allowed: \( i = 0, 2, 4, \ldots, 2020. \)
Step 2: Count even numbers from 0 to 2020.
Number of even integers = \( \frac{2020}{2} + 1 = 1011. \)
Hence, dimension of \( W = 1011. \)
Final Answer: \[ \boxed{1011} \]
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