Question

Determine if the described set is a subspace. Assume a, b, and c are real numbers. The subset of R3 consisting of vectors of the form a b c , where abc

Answer 1

This question is incomplete, the complete question is;

Determine if the described set is a subspace. Assume a, b, and c are real numbers.

The subset of R³ consisting of vectors of the form $\left[\begin{array}{ccc}a\\b\\c\end{array}\right]$ , where abc = 0

• The set is a subspace
• The set is not a subspace

Answer:

Therefore; The set is not a subspace

Step-by-step explanation:

Given the data the question;

the subset R³;

S = {  $\left[\begin{array}{ccc}a\\b\\c\end{array}\right]$ , where abc = 0 }

we know that a subset of R³ is a subspace if it stratifies the following properties;

1. it contains additive identity
2. it is closed under addition
3. it is closed under scales multiplication

Looking at the properties, we can say that it is not a subspace

As;

         u = $\left[\begin{array}{ccc}1\\1\\0\end{array}\right]$ ∈ S       and v = $\left[\begin{array}{ccc}0\\1\\1\end{array}\right]$  ∈ S

As    1×1×0=0                          0×1×1=0

But   u+v = $\left[\begin{array}{ccc}1+0\\1+1\\0+1\end{array}\right] =$ $\left[\begin{array}{ccc}1\\2\\1\end{array}\right]$  ∉  S     as 1×2×1 ≠ 0

Hence, it is not closed under addition.

Therefore; The set is not a subspace