Question

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

A. V=Rⁿ and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×nmatrix.
B. V=Pₙ and S is the subset of Pₙ consisting of those polynomials satisfying p(0)=0.
C. V=C²(I), and S is the subset of V consisting of those functions satisfying the differential equation y"−4y′+3y=0.
D. V=R³, and S is the set of vectors (x₁ , x₂ ,x₃ ) in V satisfying x₁ −5x₂+x₃=4.
E. V=Mₙ(R), and S is the subset of all nonsingular matrices.
F. 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)=4.
G. 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).

Answer 1

To find vector B, orthogonal to given vector à = 3.0Î + 4.0ç with the same magnitude, vector B could be -4.0Î + 3.0ç or 4.0Î - 3.0ç since both are orthogonal to A and have a magnitude of 5.0.

Step-by-step explanation:

Determining Orthogonal Vectors with Identical Magnitudes

If vectors A and B are two orthogonal vectors in the xy-plane with identical magnitudes, and given that vector à = 3.0Î + 4.0ç, we can find vector B. Since vector A and B are orthogonal, their dot product will be zero. The magnitude of A is √(3.0² + 4.0²) = 5.0. We need another vector with the same magnitude that is orthogonal to Ã. Vector B could be -4.0Î + 3.0ç or 4.0Î - 3.0ç because both are orthogonal to A and have the same magnitude of 5.0.

By applying the property of orthogonal vectors and their dot product being zero, we determine the components of vector B that satisfy these conditions.