True or False a T F Let A and B be two subsets of a vector s

True or False. (a) T F: Let A and B be two subsets of a vector space V such that Span{A} = Span{B}, then A = B. (b) T F: Let A be an m Times n matrix, then the solution set of Ax = 0 is a subspace of R^n. (c) T F: Let A be an m Times n matrix, then the solution set of Ax = b is a subspace of R^n. (d) T F: If the set {u. v} is linearly independent then {2u,3v} is also linearly independent. (e) T F: If three vectors in R^3 are linearly dependent, then they must lie on the same line. (f) T F: The vectors (-2,0,1), (3,2, 5), (6, -1,1), (7,0, -2) in R^3 are linearly dependent.

Solution

(a) TRUE. This comes from the definition of spanning set.

(b) TRUE. If A is a mxn matrix, then the dimension of x has to be a nx1 (Also, The null space of A is the solution set of the equation Ax = 0. The null space of an m × n matrix is in Rⁿ )

(c) TRUE (Every solution to the equation Ax = b can be written as x = p + h , where h is a solution to the system Ax = 0)

(d) TRUE;( If u, v are linearly independent, then v is not a scalar multiple of u. Therefore 3v is also not a scalar multiple of u and therefore, 3v is not a scalar multiple of 2u)

(e) TRUE: (If 2 vectors in R³ are linearly dependent, then one of these is a scalar multiple of the other. Therefore, one vector is either an elongation or a contraction of the other vector.Therefore, the two vectors lie on the same line. The samr is true for 3 vectors.)

(f) FALSE; (The RREF of the matrix having these 4 vectors as rows, has a zero row. Therefore, these are not linearly independent)