In parts ak determine whether the statement is true or false

In parts (a)-(k) determine whether the statement is true or false, and justify your answer. (a) Every subspace of a vector space is itself a vector space. (b) Every vector space is a subspace of itself. (c) Every subset of a vector space V that contains the zero vector in V is a subspace of V. (d) The set R^2 is a subspace of R^3. e) The solution set of a consistent linear system Ax = b of m equation in n unknowns is a subspace of R^n. f) The span of any finite set of vectors in a vector space is closed under addition and scalar multiplication. (g) The intersection of any two subspaces a vector space of. V is a subspace of V. (h) The union of any two subspaces of a vector space V is space. a subspace of V. (i) Two subsets of a vector space V that span the same subspace of V must be equal. (j) The set of upper triangular n times n matrices is a subspace of the vector space of all n times n matrices. (k) The polynomial x - 1, (x - 1)^2, and (x - 1)^3 span P_3.

Solution

(a) By the definition of a subspace, the statement is True.

(b) Since every set is a subset of itself, hence every vector space is a subspace of itself. The statement is true.

(c) The statement is false. The subset also needs to be closed under ‘multiplication’ and ‘addition’. e.g. {0, 1} is not a subspace of R while span{0, 1} is a subspace of R.

(d) The statement is false. Every subspace of R will have 3-vectors while R², having 2-vectors is not a subset of R³ and hence not a subspace of R³.

(e) The statement is false as every subspace of Rⁿ must have the 0 vector.

(f) By the definition of span, the statement is true.

(g) The statement is true. If both U,W are subspaces of V, then 0 U and 0 W, so that 0 UW. Further, if u,w U W, then u, w U and u,w W. Since U, W are subspaces, they are closed under addition which means that u + w U and u + w W. This implies that u + w UW. Also, if u UW , then u is in U and in W, so that cu U and cu W for any scalar c. Hence cu UW. Thus UW is a subspace of V.

(h) The statement is false. Let U = span{(1,0)} and W = span{(0,1)} . Then (1,0) U and (0,1) W, but u +w = (1,1) either U or W so that u+w U U W.

(i) The statement is false. The sets U = {(1,1)^T, (0,1)^T} and W = {(1,0)^T,(0,1)^T} both span R², but U W.

(j) The statement is true. The set of nxn upper triangular matrices , being closed under vector addition and scalar multiplication, and also having the 0 matrix, is a vector space.

(k) The statement is false. (x-1)³ = x³-3x²+3x-1, (x-1)²= x-2x+1 and x-1. The RREf of the coefficients, i.e. the RREF of the matrix A =

1

0

0

-3

1

0

3

-2

1

-1

1

-1

is B =

1

0

0

0

1

0

0

0

1

0

0

0

so that 1 span{ (x-1)³, (x-1)², (x-1)}. Hence, the set { (x-1)³, (x-1)², (x-1)} does not span P₃.

1	0	0
-3	1	0
3	-2	1
-1	1	-1