# Calculate a function using the fact that is multilinear and alternant

I'm currently working in the following excercise:

Be $$f$$ defined in $$\mathbb{R}^4 \times \mathbb{R}^4 \times \mathbb{R}^4 \times \mathbb{R}^4$$ a function, calculate it in

$$\begin{pmatrix} 1\\ 2\\ 0\\ 1 \end{pmatrix}, \begin{pmatrix} 3\\ 2\\ 4\\ 0 \end{pmatrix}, \begin{pmatrix} 0\\ 1\\ 7\\ 2 \end{pmatrix}, \begin{pmatrix} 1\\ 6\\ 3\\ 1 \end{pmatrix}$$

Using the fact that $$f$$ is multilinear and alternant

How could I calculate a function without knowing it? Also I've tried to play with the information provided but is not too clear to me how the fact that $$f$$ is multilinear and alternant helps. Thanks in advance for any hint or help and for taking the time to read my question.

• Honestly, the only thing that I could think of was showing that the vectors are linearly dependent, which would imply that the image is $0$. However, they're not as the determinant of the $4\times4$ matrix is $131$ and not $0$. So I have to ask, did you copy the values correctly? – Paulo Mourão May 23 at 23:56
• Notice that the fourth vector can be rewritten as a linear combination of the first three – Victoria M May 24 at 0:17
• How would you do that? – Paulo Mourão May 24 at 0:18

Show that $$f=\lambda\det$$ for some $$\lambda\in \mathbb{R}$$. Then compute the determinant.
• How would you figure out what $\lambda$ should be? Is there even sufficient information? – peek-a-boo May 24 at 15:43
• There isn't, $f$ is multilinear and alternate whatever is $\lambda$. The most you can do is to express the solution depending on $\lambda$. – Isao May 28 at 13:50