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}

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  • kb
    Lv 7
    7 years ago
    Favorite Answer

    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!

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