i couldnt get why vector (0.5,1) is shown as straight line in previous examples the utils function shows matrices vectors in correct way but i couldnt get this one.

It’s not plotting the vector (0.5, 1) there, right? It’s plotting what happens to the eigenvectors of that transformation, which are (1,0) and (-1,0). Think about what eigenvectors mean: what vector when you feed it through that transformation will still point in the same direction afterwards? Only one that has 0 as the y component, right? So the graph ends up not being very interesting in that case.

yea i misunderstood how the util function is working. Thank you.

Actually i got one more question

if the basis vectors like shown in the image that our matrice transformed (One of them is like diagonal) then the time we multiply by the eigenvectors how its seems being projected on x-axis if the eigenvectors doesnt change the direction by definition?

Because the eigenvectors of this transformation are (1,0) and (-1,0), which both point along the x-axis. That was the point I made in my previous reply: it is because of the specific nature of this transformation.

The previous graph is showing how the *standard* basis vectors (1,0) and (0,1) are transformed by the shear transformation.

well, still i didnt understand the difference of this example and main one. isnt 2 of them creates a parallelogram ? this example is make sense to me but i cant catch the main one

Well, that’s a graph of a different linear transformation, right? The point is there are an infinite number of different possible linear transformations. It matters which one you are dealing with. The point of the graphs is to help you visualize the properties of the particular transformation in question. They show two different graphs for each transformation: one using basis vectors that are not eigenvectors of the transformation and one in which the basis vectors are eigenvectors.

Well, i realized that i had a huge misconception with the definition of terms, i’ve just reviewed and now i have more clear mind. Thanks for ur answer and patience. This un/graded assignments really helpfull to correct and reinforce that known wrongly things