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Let $u,v,w \in \mathbb{R}^4$ as :

$u= \begin{pmatrix} 2\\ 1 \\0\\1 \end{pmatrix}$ $v= \begin{pmatrix} 1\\ 3 \\1\\-1 \end{pmatrix}$ $u= \begin{pmatrix} -1\\ 0 \\1\\2 \end{pmatrix}$

I have to find a system of equation of $Vect(u,v,w)$ but I always fail to find it...

How many equation must it have?

How to find it the fastest way?

1 Answers 1

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Guessing that the third vector you denoted is actually $w$, we find that $x \in \text{vect}(u,v,w)$ if and only if $x = \lambda_1u + \lambda_2v + \lambda_3w$ where $\lambda_i \in \mathbb{R}$. This gives you your system of equations.

To find the number of equations, you can define a matrix and look at its rank: the matrix is the following: \begin{equation} \begin{pmatrix} 2 & 1 & 0 & 1 \\ 1 & 3 & 1 & -1\\ -1 & 0 & 1 & 2 \end{pmatrix}. \end{equation} This matrix follows from $x = \lambda_1u + \lambda_2v + \lambda_3w$. If you use Gaussian Elimination, you should find that this matrix is equivalent to $$ \begin{pmatrix} 1 & 0 & 0 & 1.5\\ 0 & 1 & 0 & -2\\ 0 & 0 & 1 & 3.5 \end{pmatrix}$$ so it has rank 3.

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    I'm sorry but I don't understand how to find the number of equation then? – 2017-02-26
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    well suppose the rank is 2 (instead of 3), then it is possible to describe the space by 2 instead of 3 equations. Use Gaussian elimination to find out if there is a zero row in this matrix. – 2017-02-26
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    I never used matrix so it's a bit hard for me... How can I do it without that? – 2017-02-26
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    You have no idea what matrices are? Or you never tried to solve this using matrices? Note that matrices are just a way to denote systems of linear equations, so you could repeat the whole in equation-notation, try to eliminate variables and in this way find out wheather or not there are redundant equations... – 2017-02-26
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    I learn what is a matrix, but I don't know how to solve a system with a matrix. – 2017-02-26
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    You have not learned yet about rank of a matrix and Gaussian elimination? (Do you know how to solve a system of linear equations using matrices?) – 2017-02-26