Question

Consider the function $\mathit{F}\left(A\right)\mathbf{=}\mathit{F}\left[\overrightarrow{v}\overrightarrow{w}\right]\mathbf{=}\overrightarrow{\mathbf{v}}\mathbf{}\mathbf{\times }\mathbf{}\overrightarrow{\mathbf{w}}$from ${\mathbf{ℝ}}^{\mathbf{2}\mathbf{\times }\mathbf{2}}$ to$\mathrm{ℝ}$, the dot product of the column vectors of A.
a. Is Flinear in both columns of A? See Example 6.
b. Is F linear in both rows of A?
c. Is Falternating on the columns of A? See Example 4.

Short answer

1. Yes.
2. No.
3. No.

Step-by-step solution

Step 1: Given

Given,

(a) Is  linear in both columns of?

Yes, F is linear in both columns. Linearity in the both columns follows from the properties of the scalar product decoded by F.

Let $\overrightarrow{v}=\left(a,b\right)$and $\overrightarrow{w}=\left(c,d\right)$and let us arrange these vectors in columns. Observe that

This is linear in both columns since the dot product is linear on both left and right side.

(b) Is  linear in both rows of?

For the same reason F is linear in both columns, F fails to be linear in both rows. F decodes a dot product so it is impossible for F to yield a linear combination of bothandwhen tested for linearity.

(c)  Is F alternating on the columns of A?

Recall that for two vectorsthe following relation holds:

Since interchanging the columns doesn't change the sing of F , F is not alternating on the columns.