Matrices and Linear spaces

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1. Apr 9, 2017

Faiq

1. The problem statement, all variables and given/known data
The vectors $a_1, a_2, a_3, b_1, b_2, b_3$ are given below
$$\ a_1 = (3~ 2~ 1 ~0) ~~a_2 = (1~ 1~ 0~ 0) ~~ a_3 = (0~ 0~ 1~ 0)~~ b_1 = (3~ 2~ 0~ 2)~~ b_2 = (2 ~2~ 0~ 1)~~ b_3 = (1~ 1~ 0~ 1)$$
The subspace of $\mathbb R^4$ spanned by $a_1, a_2, a_3$ is denoted by $V_1$ and the subspace of $\mathbb R^4$ spanned by $b_1, b_2, b_3$ is denoted by $V_2$

The set of vectors which consists a zero vector and all vectors which belong to $V_1$ and $V_2$ is denoted by $W$.

Write down two linearly independent vectors which belong to $W$

3. The attempt at a solution
Can somebody please explain how to get the independent vectors? I am very aware of the definition of independent vectors. However, I cannot seem to relate the definition and use it to solve this question.

2. Apr 9, 2017

FactChecker

Hint: a1, a2, and a3 all have 0 as their 4'th coordinate, so they can not generate any vector with a nonzero 4'th coordinate.

3. Apr 9, 2017

Faiq

And $b_1, b_2, b_3$ all have 0 as their 3rd coordinate, so they cannot generate any vector with a nonzero 3rd coordinate
Thus the resulting vector should be $(p~q~0~0)$
Correct?

4. Apr 9, 2017

FactChecker

You need to find two vectors. Look for one in V2 that could not be a linear combination of the ais and look for a vector in V1 that could not be a linear combination of the bis

5. Apr 9, 2017

Faiq

a3 and b3?
And the reasoning behind the special criteria for choosing the vector is this right? If I were to choose a vector in V1 which was a linear combination of V2 vectors, then I would technically be choosing just V2 vectors and not taking into account the V1 vectors.

6. Apr 9, 2017

FactChecker

In homework, you would want to briefly explain why the two you pick are in W and are linearly independent. Use the definition of linearly independent.

7. Apr 9, 2017

Faiq

Thank you very much for your help