Let W₁ be the set: [1 0 0], [1 1 0], [1 1 1] and Let W₂ be the set: [1 0 1], [0 0 0], [0 1 0]

Question

Determine if W₁ is a basis for R³ and check the correct answer(s) below.

A. W₁ is a basis.

B. W₁ is not a basis because it does not span R³.

C. W₁ is not a basis because it is linearly dependent.

Determine if W₂ is a basis for R³ and check the correct answer(s) below.

A. W₂ is not a basis because it is linearly dependent.

B. W₂ is not a basis because it does not span R³.

C. W₂ is a basis.

Answer 1

If a[1 0 1] + b[0 0 0] + c[0 1 0] = [0 0 0], then a, b, and c don't all have to be 0. For instance, a = 0,

b = 2, and c = 0 works. So, W₂ is not a linearly independent set. Therefore, W₂ is not a basis of R³.

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If a[1 0 0] + b[1 1 0] + c[1 1 1] = [0 0 0], then

a+b+c = 0

b+c = 0

c = 1

So, a = b = c = 0. Therefore, W₁ is a set of 3 linearly independent vectors in the 3-dimensional set R³. So, W₁ is a basis of R³.