Is the subset W x y z z 1 R3 a vector subspace of R3 Exp

Is the subset W = {(x, y, z) | z = 1} R^3 a vector subspace of R^3 ? Explain why or why not.

If W is the subset of all vectors (x, y) in R^2 such that |x| = |y|, is W a vector subspace or not?

First part

(0,0,1) is one vector in W

Multiplying it by a scalar ,2 gives the vector: (0,0,2) which is not in W

HEnce, W is not a subspace

Second Part

Consider two vectors in this subset

(3,-3)

(-4,-4)

which are both in W

Adding the two vectors gives

(-1,-7) which is not in W

Hence W is not a subspace

$Is the subset W = {(x, y, z) | z = 1} R^3 a vector subspace of R^3 ? Explain why or why not. If W is the subset of all vectors (x, y) in R^2 such that |x| = |y|$

Is the subset W x y z z 1 R3 a vector subspace of R3 Exp

Solution

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