00:01
In this question, we are given a linear transformation t from r3 to r3 by t of x, y, z is equal to x plus y, y plus z, and z minus 2x.
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And here, we are asked to find the matrix of the transformation t.
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So, and the recipe and the formula is very easy, simple.
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A is equal to t of e1, t of e2, t of e3.
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Where e1, e2 and e3 are standard basis vectors for r3.
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E1 is equal to 1 00, e2 is equal to 010, and e3 is equal to 0 -01.
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Now we need to see what happens to these vectors to basis vectors under the transformation t.
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In other words, we need to apply this formula to vectors e1, e2 and e3.
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So t of 1 0 0 is equal to x plus the first entry is going to be x plus y.
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In our case, x is 1 and y is 0, so it's going to be just 1.
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The second entry is y plus z, so the sum of the last two entries and it's 0.
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And the third entry is 0 minus 2x, z minus 2x.
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In our case, it's going to be 0 minus 2.
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So it's going to be negative 2.
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Next let's find t of 0 -1 -0.
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Now x and z are equal to 0 and y is equal to 1.
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So we're going to get x plus y is going to be equal to 1.
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Y plus z will be equal to 1 and z minus 2 x is 0 because both z and x are 0.
02:23
Finally t of 0 0 is 0 is equal to now x y are 0 and z is equal to 1 x plus y is going to be 0 and y plus z is going to be 1 and z minus 2 x will be equal to 1 therefore the matrix of our transformation consists of the images of the basis vectors so basically the first column is going to be 1 0 negative 2 the second column is going to be the image of e2 which is 110 and the third column is going to be 011.
03:27
This is the matrix of our transformation.
03:32
Now, in the second part of this question, we're asked to find the image of the vector t of 1 -0 -1 -15.
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And to do that, we simply need to multiply this vector by the matrix a.
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So we're going to get...
04:16
So 1 -1 -0 times 1 -925 is going to give us 1 -1 -1.
04:23
Will be 0 then we're going to get 0 minus 1 plus 5 and the last answer is going to be negative 2 plus 0 plus 1 so the the image is going to be 0 4 negative 1 now we're asked to find the image of t inverse of 0 4 3 to do that we basically need to do a inverse times 0 4 3 but we don't have a inverse yet, so we have to find it...