Let V be the vector space of polynomials with degree not exceeding two. Let S = {1, x -1, x2 - 2x + 1} be a subset of V. Then which of the following(s) is/are true ?a)S form a basis for Vb)S not form a basis for Vc)Co- ordinate vector of (2x2 - 5x + 6) relative to S is (3, -1, 2)d)We cannot find out Co - ordinate vector of (2x2 - 5x + 6) relative to S.Correct answer is option 'A,C'. Can you explain this answer?

Question

Answer

We know dim V = 3 and S is a linearly independent subset of V which has 3 elements = S form a basis for V

Now take (2x²- 5x + 6) = a(1) + b ( x - 1) + c(x² - 2x + 1) = 3(1) + (-1) ( x -1)+ 2(x² - 2x + 1) = 2x²- 5x + 6 = O ption (C) is correct

Answer

Understanding the Vector Space V
The vector space $ V $ consists of polynomials of degree at most 2, which can be expressed in the form:
$$
p(x) = ax^2 + bx + c
$$
where $ a, b, c $ are real coefficients.

Analyzing the Set S
The set $ S = \{1, x - 1, x^2 - 2x + 1\} $ contains three polynomials. To determine whether $ S $ forms a basis for $ V $, we need to check two things:
1. **Linear Independence**
2. **Spanning the Space**

1. Linear Independence
To check if the polynomials in $ S $ are linearly independent, we can express the polynomial $ x^2 - 2x + 1 $ as:
$$
x^2 - 2x + 1 = (x - 1)^2
$$
Thus, $ S $ can be rewritten as:
$$
S = \{1, x - 1, (x - 1)^2\}
$$
Notice that $ (x - 1) $ and its square are not linearly independent since one can be expressed in terms of the other. Therefore, $ S $ is not linearly independent.

Conclusion on Basis
Since the polynomials in $ S $ are not linearly independent, they do not form a basis for $ V $. Thus, option (b) is true.

Finding the Coordinates
To find the coordinates of the polynomial $ 2x^2 - 5x + 6 $ relative to $ S $, we must express it as a linear combination of the elements in $ S $:
$$
2x^2 - 5x + 6 = a(1) + b(x - 1) + c(x^2 - 2x + 1)
$$
Solving for $ a, b, c $ will yield the coordinates.
The statement (c) claims that the coordinates are (3, -1, 2). This requires verification through substitution. Therefore, we can find the coordinates, making option (d) false.

Final Answers
- **True:** Option (a) is false, option (c) is true.
- **False:** Option (b) is true, option (d) is false.

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