It is extremely important to observe that if c element V the

It is extremely important to observe that if c element V then all elements the linear equation c middot x = 0. Thus, there is an intimate relation between and Cartesian equations defining the subspace V. We will explore and exploit, more fully in the next few sections. It will be useful for us to make the following definition. Definition. Let V and W be subspaces of R^n. We say V and W are orthogonal if every element of V is orthogonal to every element of W, i.e., if v middot w = 0 for every v element V and every w element W If V = W or W = V, then clearly V and W are orthogonal the other hand, if V and W are orthogonal subspaces of R^n, then certainly V W (See Exercise 12.) Of course, W need not be equal to V example, V to be the x_1-axis and W to be the x_2-axis in R^3. Then V which contains W and more. It is natural, however, to ask the following W = V, must V = W ? We will return to this shortly. Which of the following are subspaces? Justify your answer in each case {x element R^2: x_1 + x_2 = 1} {x element R^3 x = [a b a + b] for some a, b element R} {x element R^3: x_1 + 2x_2

Solution

Recall a subset S of a vector space is a subspace iff the following are satisfied:

x,y in S implies x+y is also in S (S is closed under addition)

x in S and c a scalar implies cs is in S . (S is closed under scalar multiplication)

In particular, taking c =0, we note that the 0 vector must belong to S.(if S is a subspace)

1d) In this case the 0 vector (0,0,0) is not in the subset S. So NOT A SUBSPACE

NOTE: (only ) b and g are subspaces in this problem: