Linear Subspace Dimension
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Please provide solutions to these two questions (attached).
Please show how the subspace satisfies both addition & scalar multiplication!
In each of the following exercises 8-17, we will denote by S the set of all vectors x = (x1, x2, x3) E R3 whose coordinates satisfy the given condition. In each case determine whether the given set S is a linear subspace of R3 and if so, compute its dimension.
16. x1 = x2 = x3
15. x2 = 2x1 and x3 = 3x1
Partial solution to 15:
{x = (x1, x2, x3) E R3: x2 = 2x1, x3 = 3x1}
So x = (x1, 2x1, 3x1)
= x1(1, 2, 3)
So that shows that the subspace is closed under scalar multiplication. And it obviously contains the zero vector: (0, 0, 0). What about addition though?---
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Solution Summary
The solution assists with answering a question that is based on the linear subspace dimension.
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About your partial solution:
It is OK.
It is closed under addition too:
a(1,2,3) + b(1,2,3) = (a+b)(1,2,3)
Perhaps you were ...
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