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```MATH 240 –Answers to T/F textbook questions
Chapter 5
Section 5.2:
1. Technically, false, since if the transformation is invertible it would map some lines to single points.
But it is true that invertible transformations map lines to lines, and all transformations map lines
either to lines or to points
2. Again, technically false — certain reflections, stretches and shears correspond to elementary
matrices, but not all of them. For example the matrix
that?stretches
the line that makes an angle of
7
3
3
π {3 with the positive x-axis by a factor of 4 is 3 ?4 4 13 which is not an elementary matrix.
4 3
4
But the matrices that stretch, shear and reflect around the coordinate axes are elementary matrices.
3. False – even for the ones along the coordinate axes. The two shears have to be along the same
axis.
4. True
5. False
6. False (they have to stretch the same direction)
1.
2.
3.
4.
5.
6.
Section 5.3:
False (it’s dim V )
False (Since dim P4 5, this map could be one-to-one)
False (this is ker T )
False (it’s a subspace of W )
True
True
Section 5.4:
1. False (since dim M32 ¡ dim P3 )
2. True (both have dimension 6)
3. True
4. True
5. True
6. True
7. False (not one-to-one because the derivative of a constant is zero)
8. True
9. True
10. False (only if the dimension of the vector space is finite)
11. False (if dim V1 1, dim V2 20 and dim V3 2, then T2 can be onto but T2 T1 cannot be)
12. False (the kernel is at least five-dimensional but for instance the dimension of the kernel of the
zero map will be 8).
Section 5.6:
1. False (v must be nonzero)
2. True
2
3.
4.
5.
6.
7.
8.
9.
True
False
True
True
False (they have to correspond to the same eigenvalue)
False (unless the matrix is real)
True
1.
2.
3.
4.
5.
6.
7.
Section 5.7:
True
True
True
True
True
False (not necessarily, e.g., the identity matrix)
True
1.
2.
3.
4.
5.
6.
7.
8.
Section 5.8:
True
True (same eigenvectors, eigenvalues are reciprocals of those of A)
False (one might be diagonalizable, the other not)
False (n linearly independent eigenvectors)
True
True
True (A I 1 AI)
True (the sum of even numbers can’t be odd, so A doesn’t have n independent eigenvectors)
1.
2.
3.
4.
5.
Section 5.9:
True
False (should be 21 A2 )
False (eAt is always invertible, even if A is not)
True
False (should be pSDS 1 qk SDk S 1 )
Chapter 6
1.
2.
3.
4.
5.
6.
7.
Section 6.1:
True
False (that can’t happen — either W is always zero or never zero on I)
False (unless the operators have constant coefficients)
True
True
True (assuming the differential equation is linear)
True
3
1.
2.
3.
4.
5.
6.
7.
8.
Section 6.2:
False (the equation always has n independent solutions)
False (constant-coefficient linear ones do, though)
True
True
False (the equation only has order 5, so not 6 constants – x is not a solution)
True
True
False
Section 6.3:
1. False (but A1 pDqA2 pDq does)
2. False (it’s ApDq Dk 1 )
3. True
4. Strictly speaking, false (since ApDq could also work, but if you restrict to operators with leading
term Dk then true)
5. False (think of F pxq xex and let A1 A2 D 1)
6. False (just yp A2 x2 A3 x3 )
7. False (need A1 x3 A2 x4 A3 x5 A4 x6 A5 x7 , i.e., multiply a polynomial of degree 4 by x3 )
8. False (don’t need to multiply by x)
1.
2.
3.
4.
5.
6.
Section 6.4:
True
False (also need a term without the factor of x)
False (should be A0 xeix )
True
False (should be pA0 x A1 x2 qep2 5iqx )
True
1.
2.
3.
4.
5.
6.
7.
8.
9.
Section 6.5:
True
False (square root of that)
True
True (unless c 0)
False
True (assuming they’re talking about circular frequency in both instances of the word)
True
a
False (m is in the numerator of T 2π m{k)
True
Section 6.7:
1. True
2. True
3. False (you can add constants to them, at least)
4
```
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