Question

Determine whether the given set s is a subspace of the vector space v.

a. v=mn(r), and s is the subset of all upper triangular matrices.


b. v is the vector space of all real-valued functions defined on the interval (−∞,∞), and s is the subset of v consisting of those functions satisfying f(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=c1(r), and s is the subset of v consisting of those functions satisfying f′(0)≥0.


e. v=r2, and s is the set of all vectors (x1,x2) in v satisfying 5x1+6x2=0. f. v=p5, and s is the subset of p5 consisting of those polynomials satisfying p(1)>p(0). g. v=mn(r), and s is the subset of all nonsingular matrices.

Answer 1

a) Yes, s is a subspace of v because the three conditions for a subspace are satisfied. b) Yes, s is a subspace of v because the three conditions for a subspace are satisfied. c) Yes, s is a subspace of v because the three conditions for a subspace are satisfied.

Step-by-step explanation:

a. v=mn(r) is the vector space of all m x n matrices with entries in the real numbers. In this case, s is a subspace of v if it satisfies the three conditions: (1) the zero vector is in s, (2) s is closed under vector addition, and (3) s is closed under scalar multiplication. Since the zero matrix is an upper triangular matrix, it satisfies the first condition. If two upper triangular matrices are added, the result will also be an upper triangular matrix, satisfying the second condition. And if an upper triangular matrix is multiplied by a scalar, the result is still an upper triangular matrix, satisfying the third condition. Therefore, s is a subspace of v.

b. In this case, v is the vector space of all real-valued functions defined on the interval (-∞, ∞). The subset s consists of functions satisfying f(0) = 0. Similar to part a, we need to check if s satisfies the three conditions to be considered a subspace. The zero function satisfies f(0) = 0, so it satisfies the first condition. If two functions in s are added, the sum will also have f(0) = 0, satisfying the second condition. And if a function in s is multiplied by a scalar, the result will still have f(0) = 0, satisfying the third condition. Therefore, s is a subspace of v.

c. In this case, v is the vector space of all second-order linear homogeneous differential equations. The subset s consists of functions satisfying y'' - 4y' + 3y = 0. To be considered a subspace, s would need to satisfy the three conditions: (1) the zero element is in s, (2) s is closed under addition, and (3) s is closed under scalar multiplication. The zero function satisfies the differential equation, so it satisfies the first condition. If two functions in s are added, the sum will also satisfy the differential equation, satisfying the second condition. And if a function in s is multiplied by a scalar, the result will still satisfy the differential equation, satisfying the third condition. Therefore, s is a subspace of v.