Determine whether the given set S is a subspace of the vector space V. A. V={ R}

Question

Determine whether the given set S is a subspace of the vector space V. A. V={ R}^3, and S is the set of vectors (x_1,x_2,x_3) in V satisfying x_1 - 5 x_2 + x_3 = 4. 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=P_3, and S is the subset of P_3 consisting of all polynomials of the form p(x)=x^2+c. D. V=P_4, and S is the subset of P_4 consisting of all polynomials of the form p(x)=ax^3+bx. E. V=M_n({R}), and S is the subset of all upper triangular matrices. F. V=C^2(I), and S is the subset of V consisting of those functions satisfying the differential equation y''-4y'+3y=0. G. V=M_n(R}), and S is the subset of all n imes n matrices with det(A)=0.

Explanation / Answer

y''-4y'+3y=0

m^-4m+3=0

m^2-3m-m+3=0

m(m-3)+(m-3)=0

m=-1,m=3

the general solution is y(x)=c1e^x+c2e^3x

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