Need help with vector space multiple choice

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SUMMARY

The discussion focuses on determining whether specific subsets S of various vector spaces V qualify as subspaces. The criteria for S to be a subspace include closure under addition and scalar multiplication. The examples provided include polynomial spaces (P5, P4), function spaces (C2(I)), and matrix spaces (Mn(ℝ)). Notably, example G illustrates a specific linear condition, while example D highlights the importance of the determinant in assessing subspace properties.

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  • Understanding of vector spaces and subspaces
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  • Knowledge of linear algebra concepts, particularly determinants
  • Basic understanding of differential equations and function spaces
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Students and professionals in mathematics, particularly those studying linear algebra, as well as educators looking to deepen their understanding of vector space properties and subspace criteria.

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Determine whether the given set S is a subspace of the vector space V.

A. V=P5, and S is the subset of P5 consisting of those polynomials satisfying p(1)>p(0).
B. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=f(b).
C. V=ℝn, and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.
D. V=Mn(ℝ), and S is the subset of all n×n matrices with det(A)=0.
E. V=P4, and S is the subset of P4 consisting of all polynomials of the form p(x)=ax3+bx.
F. V=C2(I), and S is the subset of V consisting of those functions satisfying the differential equation yʺ=0.
G. V=ℝ3, and S is the set of vectors (x1,x2,x3) in V satisfying x1-9x2+x3=8.
 
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S is a subspace of V if, (1) S is closed under addition (given two elements a and b in S, a+b is in S) and (2)S is closed under scalar multiplication (given x in S, and a scalar c, cx is in S).

For each question you should see if the set satisfies these two criteria. If it doesn't then find a counterexample.

For example problem D, does detA = 0 = detB mean that det(A+B) = 0?
 

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