Linear Algebra, 2020a

360 Chapter Four. Determinants
(a) A (b) A 2 (c) A −2
̌ 1.13 Consider the linear transformation t of R 3 represented with respect to the
standard bases by this matrix.
⎛
⎞
1 0 −1
⎝ 3 1 1 ⎠
−1 0 3
(a) Compute the determinant of the matrix. Does the transformation preserve
orientation or reverse it?
(b) Find the size of the box defined by these vectors. What is its orientation?
⎛ ⎞
1
⎝−1⎠
2
⎛
⎝
2
0
−1
⎞
⎠
⎛ ⎞
1
⎝1⎠
0
(c) Find the images under t of the vectors in the prior item and find the size of
the box that they define. What is the orientation?
1.14 By what factor does each transformation change the size of boxes?
⎛ ⎞ ⎛
( ( ) ( ( ) x x − y
x 2x x 3x − y
(a) ↦→ (b) ↦→
(c) ⎝ ⎠ ↦→ ⎝
y)
3y y)
−2x + y
y
z
x + y + z
y − 2z
1.15 What is the area of the image of the rectangle [2..4] × [2..5] under the action of
this matrix? ( 2
) 3
4 −1
1.16 If t: R 3 → R 3 changes volumes by a factor of 7 and s: R 3 → R 3 changes volumes
by a factor of 3/2 then by what factor will their composition changes volumes?
1.17 In what way does the definition of a box differ from the definition of a span?
1.18 Does |TS| = |ST|? |T(SP)| = |(TS)P|?
1.19 Show that there are no 2×2 matrices A and B satisfying these.
( ) ( )
1 −1
2 1
AB =
BA =
2 0
1 1
1.20 (a) Suppose that |A| = 3 and that |B| = 2. Find |A 2 · B T · B −2 · A T |.
(b) Assume that |A| = 0. Prove that |6A 3 + 5A 2 + 2A| = 0.
̌ 1.21 Let T be the matrix representing (with respect to the standard bases) the map
that rotates plane vectors counterclockwise through θ radians. By what factor does
T change sizes?
̌ 1.22 Must a transformation t: R 2 → R 2 that preserves areas also preserve lengths?
1.23 What is the volume of a parallelepiped in R 3 bounded by a linearly dependent
set?
̌ 1.24 Find the area of the triangle in R 3 with endpoints (1, 2, 1), (3, −1, 4), and
(2, 2, 2). (This asks for area, not volume. The triangle defines a plane; what is the
area of the triangle in that plane?)
1.25 An alternate proof of Theorem 1.5 uses the definition of determinant functions.
(a) Note that the vectors forming S make a linearly dependent set if and only if
|S| = 0, and check that the result holds in this case.
⎞
⎠