MTH017 Linear Algebra for Maths
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MTH017 Linear Algebra for Maths
VECTOR SPACES, SUBSPACES, NULL SPACE, COLUMN SPACE AND BASES
Question 1. For each of the following subsets in some vector spaces, determine whether it is a subspace. Explain why. If the subset is a subspace, find a basis for the subspace and explain why the set you find is a basis.
a . H = ((s -t, 2s, s + t) | s, t e R} c R3 .
b. Let M2×2 consist of all the 2 × 2 matrices. M2×2 with the matrix addition and scalar
multiple operations forms a vector space. Consider the subset U = (X e M2×2 | AX + XA = O} with
A = 
!.
c ● K = (at2 + bt + c | a ≥ 0, b, c e R} c P2 .
d ● S = (p e P2 | p(1) = 0}.
Question 2 ● For each of the following sets of vectors, determine whether the set is linearly independent and find a basis for the subspace spanned by the set.
a. X = (1 - t2 , 1 + t2 , 3t + 5t2}.
2
b · (v1 =
'-5' , v2 = '-6' , v3 = '2'}.
Question 4. Is S = (A e M3×3 | A = AT } a subspace of M3×3? If yes, find a basis for it.
Question 5. Some subspaces H1 , H2 of a vector space V satisfy H1 n H2 = (0}. Show that a. h1 e H1 and h2 e H2 add up to zero, i.e., h1 + h2 = 0 if and only if h1 = h2 = 0.
b. if B1 is a basis for H1 and B2 for H2 , then B1 u B2 is a basis for H1 + H2 .
2021-12-03