For each of the statemnts decide whether it is true (T) or false (F). Substantia
ID: 3039396 • Letter: F
Question
For each of the statemnts decide whether it is true (T) or false (F). Substantiate your decision.
a) If a subset of Rn contains the zero vector, then it is a subspace of Rn.
b) If a linear manifold in Rn contains the zero-vector, then it is a subspace of Rn.
c) The spanning set of 4 vectors in R6 is a 4-dimensional subspace in R6.
d) If A is a singular matrix, then Null(A) contains infinitely many vectors.
e) If A is an m x n matrix, and rank(A) = n - 2, then Null(A) is 2-dimensional.
f) A vector from Null(A) is orthogonal to any vector from Row(A)
g) If A is a matrix size 5 by 6, and rank(A) = 3, then nullity(A) = 3.
h) The Column Space of a singular n x n matrix is n-dimensional.
Explanation / Answer
a) If a subset of Rn contains the zero vector, then it is a subspace of Rn.
FALSE. The subset V = { (1,0,0,0,…,0),(0,1,0,…,0),(0,0,0,…,0)} is not a subspace of Rn as (1,1,0,…,.0)
b) If a linear manifold in Rn contains the zero-vector, then it is a subspace of Rn.
TRUE. By definition, a linear manifold ( Ref: Halmos and Conway) means a closed linear subspace.
c). The spanning set of 4 vectors in R6 is a 4-dimensional subspace in R6.
FALSE. It has not been stated that the given set of four vectors is linearly independent or not. Unless it is linearly independent, the spanning set of these 4 vectors cannot be a FOUR-dimensional subspace in R6.
d). If A is a singular matrix, then Null(A) contains infinitely many vectors.
TRUE. Null(A) is the set of solutions to the equation AX = 0. If A is singular ( non-invertible), then then there is a nonzero X such that AX= 0.This means that kX = 0 for any arbitrary k so that there will be an infinite number of solutions.
e) If A is an m x n matrix, and rank(A) = n - 2, then Null(A) is 2-dimensional.
TRUE. By the dimension (Rank-nullity theorem), the rank of A + the nullity of A is equal the number of columns of A.
f). A vector from Null(A) is orthogonal to any vector from Row(A).
TRUE. The nullspace of a matrix is the orthogonal complement of its row space (fundamental theorem of linear algebra).
g) If A is a matrix size 5 by 6, and rank(A) = 3, then nullity(A) = 3.
TRUE. By the dimension (Rank-nullity theorem), the rank of A + the nullity of A is equal the number of columns of A.
h). The Column Space of a singular n x n matrix is n-dimensional.
FALSE. If A has a column of zeros, then A is singular and Col(A) can, at best, be n-1 dimensional.
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