sis asked in Science & MathematicsMathematics · 7 years ago

# Q2) Determine which of the following subsets of R^n are in fact subspaces of R^n (n>2)?

a) {X│x1 x2=0}

b) {X│AX=b,A_(m×n )≠0 and b_(m×n )≠0}

c) {X│x_i≥0}

d) { X ∑_(j=1)^n▒x_j =1}

Relevance
• kb
Lv 7
7 years ago

a) Not a subspace.

Counterexample (n = 2): (1, 0) and (0, 1) are in the subset, but their sum (1, 1) is not.

b) Not a subspace, since 0 is not in the space.

c) Not a subspace, since it is not closed under scalar multiplication.

For instance, although (1, ..., 1) is in the subset, -1 * (1, ..., 1) = (-1, ..., -1) is not.

d) Not a subspace, since it is not closed under scalar multiplication.

For instance, although (0, ..., 0, 1) is in the subset, 2(0, ..., 0, 1) = (0, ..., 0, 2) is not.

I hope this helps!