T(x¹) = 5x¹ + 3x³, T(x³) = −4x¹ + (-5)x³, T(x²)=-3x² + -2x² +-2x, T(x¹) = 5x¹ + 5x³ + 5x² + 2x, and T(1) = 0x¹ + 0x³ + (0)x² + 3x +3. For each of the following subspaces, answer "1" if it is T-invariant and "0" otherwise. The subspace of P with ordered basis {2x¹ + 2x³ + 2x²+2x+2}. The subspace of P, with ordered basis 4
T(x¹) = 5x¹ + 3x³, T(x³) = −4x¹ + (-5)x³, T(x²)=-3x² + -2x² +-2x, T(x¹) = 5x¹ + 5x³ + 5x² + 2x, and T(1) = 0x¹ + 0x³ + (0)x² + 3x +3. For each of the following subspaces, answer "1" if it is T-invariant and "0" otherwise. The subspace of P with ordered basis {2x¹ + 2x³ + 2x²+2x+2}. The subspace of P, with ordered basis 4
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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Transcribed Image Text:Consider linear transformation T: P₁ → P₁ given by
T(x¹) = 5x¹ + 3x³,
T(x³) = −4x¹ + (-5)x³,
T(x²) = −3x¹ +
-2x² +-2x,
T(x¹) = 5x¹ + 5x³ + 5x² + 2x, and
T(1) = 0x¹ + 0x²³ + (0)x² + 3x +3.
For each of the following subspaces, answer "1" if it is T-invariant and "0" otherwise.
The subspace of P4 with ordered basis {2x¹ + 2x³ + 2x²+2x+2}.
The subspace of P4 with ordered basis {x¹, x¹+x³}.
The subspace of P₁ with ordered basis {2x¹ + 2x³ + x², -x¹ - x³ + x²}.
The subspace of P4 with ordered basis {2x¹ + 2x³ + x² +x+1, x²+x+1, 3x¹+x² + x +1}.
The subspace of P with ordered basis {x¹, x¹+x³+x²+x}.
The subspace of P with ordered basis
{x¹, x² + x³, x¹+x³+x²,
x¹+x³+x²+x, x¹+x³+x²+x+1}.
Let U denote the 2-dimensional T-invariant subspace from above, and let B denote the given ordered basis.
Using Ty to denote the restriction of T to U.
Ms(T|u)=
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