Determine if the given set W is a subspace of the given vector space V. If so, find a possible basis for W and give the dimension of W. (a) V = M2x2 (R), W = (b) V = {( M3x3 (R), W: = x² - y² x-y x+y xy matrices with real coefficients : x, y R = = -A}, the set of all skew-symmetric 3 × 3 {A E M3x3(R): AT
Determine if the given set W is a subspace of the given vector space V. If so, find a possible basis for W and give the dimension of W. (a) V = M2x2 (R), W = (b) V = {( M3x3 (R), W: = x² - y² x-y x+y xy matrices with real coefficients : x, y R = = -A}, the set of all skew-symmetric 3 × 3 {A E M3x3(R): AT
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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solve parts a and b only
Transcribed Image Text:Determine if the given set W is a subspace of the given vector space V. If so, find a possible
basis for W and give the dimension of W.
(a) V = M₂x2 (R), W =
(b) V = M3×3 (R), W
{ ( 2² = 1² x + y) : x₁
x - y
=
: x, y ER
{A € M3x3(R) : AT = -A}, the set of all skew-symmetric 3 × 3
matrices with real coefficients
(c) V = R₂(X), W = {(2a − 3b + 1) + (−2a + 5b)X + (2a + b)X²: a, b = R}
(d) V = R4(X),
W = {ao + a₁X + a₂X² + a3X³ + α₁X¹ ao, a₁,..., an € R, ao + a₁ + a₂ + a3 + a₁ = 0}
:
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