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Help proving a subset is a subspace

  • Thread starter paulrb
  • Start date
  • #1
18
1

Homework Statement



Prove that the set of all 3-vectors orthogonal to [1, -1, 4] forms a subspace of R^3.

Homework Equations



Orthogonal means dot product is 0.

The Attempt at a Solution



I know the vectors in this subspace are of the form
[a,b,c] where a - b + 4c = 0.
However I don't know how to use this to show there is closure under vector addition and scalar multiplication.
 

Answers and Replies

  • #2
dx
Homework Helper
Gold Member
2,011
18
Let v_1 and v_2 be two vectors orthogonal to the given vector (call it a), i.e.

[tex] v_1 \cdot a = 0 [/tex]
[tex] v_2 \cdot a = 0 [/tex]

Now, using these, all you have to do is show that

[tex] (v_1 + v_2) \cdot a = 0[/tex]

HINT: Use distributivity of the dot product.
 

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