(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Determine whether the subset W of the vector space V is a subspace of V.

Let V = L(Q^{4}) (the set of linear transformations from rational numbers with 4 coordinates to rational numbers with 4 coordinates).

Let W = { T in V = L(Q^{4}) | { (1,0,1,0) , (0,1,0,-1) } is contained in N(T) }

where N(T) is the nullspace of T

2. Relevant equations

Subspace test involves checking if the zero vector is in W, whether W is closed under addition, and whether W is closed under scalar multiplication.

3. The attempt at a solution

1. Check if zero vector is in W

From definition, N(T): {x in V | T(1,0,1,0) = 0 , T(0,1,0,-1) = 0 }

So the zero vector is in W

2. Check if closed under addition

From linearity of T, T(x+y) = T(x) + T(y)

where x = (a, b, c, d) in W

y = (p, q, r, s) in W

3. Check if closed under scalar multiplication

From linearity of T, T(cx) = cT(x)

where x = (a, b, c, d) in W

So W is a subspace of V.

Is it enough to use the linearity of T to check for closure under addition and multiplication? Is there something else I should include in my solution? I'm unsure if I'm supposed to do the subspace test on the actual vectors x, y, etc. or if I'm supposed to test the transformations, T(x), T(y), etc. For example, should closure under addition be asking whether x + y is in W or if T(x) + T(y) is in W?

Thanks in advance!

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# Homework Help: Subspace test involving linear transformations

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