Question

Let T : \(P_2(\R) → P_4(\R)\) be the linear transformation given by T(p(x)) = p(x²). Then the rank of T is equal to __________.

Answer 1

Correct Answer: 3

To determine the rank of the linear transformation \( T: P_2(\mathbb{R}) \to P_4(\mathbb{R}) \) defined by \( T(p(x)) = p(x^2) \), let's first analyze the spaces involved:

• Domain: \( P_2(\mathbb{R}) \) consists of all polynomials of degree ≤ 2. A general polynomial in this space is \( p(x) = ax^2 + bx + c \).
• Codomain: \( P_4(\mathbb{R}) \) consists of all polynomials of degree ≤ 4.

Applying the transformation:

• \( T(ax^2 + bx + c) = a(x^2)^2 + b(x^2) + c = ax^4 + bx^2 + c \).

The image of each polynomial \( ax^2 + bx + c \) is of the form \( ax^4 + bx^2 + c \). Notice that this shows the transformed polynomials are linear combinations of \( x^4, x^2, \) and \( 1 \).

Thus, the image of \( T \) is spanned by \( \{x^4, x^2, 1\} \) and has dimension 3, indicating the rank of \( T \). Now, let us verify the rank:

• The linearly independent set is \(\{x^4, x^2, 1\}\).
• The rank of a linear transformation is the dimension of the image, so rank \( T = 3 \).

Since the problem provides a range of (3,3), our calculated rank of 3 fits perfectly within this range. Thus, the rank of \( T \) is confirmed to be 3.