Linear transformation of a matrix

[mikewill54]

Hi
Im working through some old exam problems in preparation for an exam. And I’ve got no idea how to approach the following question, I’m hoping someone can help
Thanks
Mike

 

[Deleted member 4993]

Hi
Im working through some old exam problems in preparation for an exam. And I’ve got no idea how to approach the following question, I’m hoping someone can help
Thanks
MikeView attachment 28537

What do you "think" is the correct response and Why?

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Please share your work/thoughts about this problem.

 

[Otis]

I’ve got no idea how to approach [this question]

Hi Mike. I'd sketch a couple vectors (added) and apply the transformation (i.e., do the matrix multiplication with each vector, to obtain the new vectors) and sketch that too. Then I'd compare the resulting triangles' shapes, positions and orientations.

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[mikewill54]

Hi
Thanks for the replies… what do you mean by matrices (added) I can’t find my notes. I’ve just managed to dig out some old exam papers. Vectors and matrices are the only things that I’m having trouble with. I know the top left number gives an x scaling, top right an x shearing. Then the bottom left gives a y shering and bottom right gives a y scaling.
Regards
Mike

 

[Harry_the_cat]

I would sketch the unit square, ie the square joining A(0,0), B(1,0), C(1,1) and D(0,1). Apply the matrix to each of the position vectors of this square and see what the resulting figure A'B'C'D' is.

 

[mikewill54]

Hi
How would I describe this from the options. There’s a reflection in the y-axis but it’s also been stretched in the x axis by 4 and sheared from what I can tell. Like I said I’ve lost the notes on these so I’m struggling a bit

 

[Harry_the_cat]

It's not simply a reflection, rotation or flattening, so you can cross out the first three options.
Label your original square ABCD. Label the image of A as A', etc. Then you will see if orientation is preserved or reversed.

 

[Harry_the_cat]

Also, your tranformation matrix should be on the left of your position vector, otherwise the multiplication is not possible.

 

[Harry_the_cat]

Another thing... the sign of the determinant of the matrix will tell you if the orientation is preserved or reversed.
If det > 0, orientation is preserved.
If det < 0, orientation is reversed.

 

[mikewill54]

Excellent thanks… I’ll work through all these now

 

[mikewill54]

Isn’t this just a flattening?

 

[Otis]

what do you mean by matrices (added)

Sketching is a graphical approach. An easy, arbitrary sketch could be a pair of 2D vectors added: that would graph as a second arrow positioned with its tail at the head of the first arrow. The transformed sketch provides an opportunity to compare the resulting triangles.

Sketching the cat's unit square and its transformation is an alternative to what I had in mind.

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[Otis]

Isn’t this just a flattening?

Check your arithmetic for points C and D, Mike.

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[mikewill54]

Thanks for the heads up on the points… I’ve still got no idea what kind of transformation it is or how I interpret it. Any help is greatly appreciated

 

[mikewill54]

is there an example online somewhere of what you mean by this
“I'd sketch a couple vectors (added) and apply the transformation”
Like I said I’ve lost the notes on matrices and I can’t find much online about this.
Thanks

 

[Otis]

is there an example online somewhere of what you mean by this “I'd sketch a couple vectors (added) and apply the transformation”

Yes, and it's basically what you've already done with the cat's unit square (graphing it, expressing the sides as vectors, applying matrix/vector multiplication to obtain the transformed vectors' components and graphing them).

If your request is about adding a pair of 2D vectors graphically (by positioning representative arrows head-to-tail), where the sum is a third vector that completes a triangle, then you may google keywords like how to add 2D vectors graphically. I know the linear algebra section at both khan academy and mathispower4u contain a number of videos on that topic.

I’ve lost the notes on matrices and I can’t find much online

For general lessons about linear transformations, such as those mentioned in your practice exercise, try googling keywords linear transformation examples matrix vector reflection rotation shearing orientation. You may need to peruse several sites, to cover everything; that's been my experience, when searching for well-presented information online covering a broad topic.

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[Otis]

You may find this page helpful, as a basic intro.

Also, have you calculated the determinant of the transformation matrix, as suggested by the cat in post #9? I think that narrows the exercise choices to one.

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[mikewill54]

Hi
Thanks for the help the determinant is -2 so it’s D cause the orientation is reversed
Thanks
Mike