Let S = {a+bx+cx²: {V₁, V2, V3} and T and = {w₁, W2, W3} be two bases in the vector space P₂ a, b, c = R}, where v₁ = 1 + x + 3x², v₂ = −2 − 2x - 7x², v3 = 1 + 2x + 6x² = W1 1 + x - x², W₂ 1 + 2x², W3 = = = 3– x+5x2. Find the transition matrix from T to S. Let v = 201 (C₁, C2, C3) so that v = C₁w₁ + C₂W2 + C3W3. Hint: For the second part, you need also to find the transition matrix from S to T. = - 3v2 + v3, find the coefficients
Let S = {a+bx+cx²: {V₁, V2, V3} and T and = {w₁, W2, W3} be two bases in the vector space P₂ a, b, c = R}, where v₁ = 1 + x + 3x², v₂ = −2 − 2x - 7x², v3 = 1 + 2x + 6x² = W1 1 + x - x², W₂ 1 + 2x², W3 = = = 3– x+5x2. Find the transition matrix from T to S. Let v = 201 (C₁, C2, C3) so that v = C₁w₁ + C₂W2 + C3W3. Hint: For the second part, you need also to find the transition matrix from S to T. = - 3v2 + v3, find the coefficients
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let \( S = \{v_1, v_2, v_3\} \) and \( T = \{w_1, w_2, w_3\} \) be two bases in the vector space \( P_2 = \{a + bx + cx^2 : a, b, c \in \mathbb{R}\} \), where
\[
v_1 = 1 + x + 3x^2, \, v_2 = -2 - 2x - 7x^2, \, v_3 = 1 + 2x + 6x^2
\]
and
\[
w_1 = 1 + x - x^2, \, w_2 = 1 + 2x^2, \, w_3 = 3 - x + 5x^2.
\]
Find the transition matrix from \( T \) to \( S \). Let \( v = 2v_1 - 3v_2 + v_3 \), find the coefficients \( (c_1, c_2, c_3) \) so that \( v = c_1w_1 + c_2w_2 + c_3w_3 \).
*Hint: For the second part, you need also to find the transition matrix from \( S \) to \( T \).*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc52b3a7d-5814-4bdd-8c26-96eede61577b%2Fe323fd79-e4d4-4bbb-b677-2e230d5d0963%2F9ryrl1_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( S = \{v_1, v_2, v_3\} \) and \( T = \{w_1, w_2, w_3\} \) be two bases in the vector space \( P_2 = \{a + bx + cx^2 : a, b, c \in \mathbb{R}\} \), where
\[
v_1 = 1 + x + 3x^2, \, v_2 = -2 - 2x - 7x^2, \, v_3 = 1 + 2x + 6x^2
\]
and
\[
w_1 = 1 + x - x^2, \, w_2 = 1 + 2x^2, \, w_3 = 3 - x + 5x^2.
\]
Find the transition matrix from \( T \) to \( S \). Let \( v = 2v_1 - 3v_2 + v_3 \), find the coefficients \( (c_1, c_2, c_3) \) so that \( v = c_1w_1 + c_2w_2 + c_3w_3 \).
*Hint: For the second part, you need also to find the transition matrix from \( S \) to \( T \).*
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