HW 3
31
Show directly from the denition that the function L : R3 R2 given by
L(x) =
2x1 x2 + 4x3
x1 6x3
is linear.
Solution:
First, the input vector x is an element of R3 (according to the notation
3
2
L : R R ), so it is of the form x = [x1 , x2 , x3 ]T
HW 4
38
Let L : R3 R2 be the linear function given by
L(x) =
2x1 x2 + 5x3
.
7x2 + 4x3
(a) Find the matrix of L.
(b) Use part (a) to nd L([3, 2, 1]T ).
(c) Find L([3, 2, 1]T ) directly from the formula for L and verify that it agrees with the
answer to par
HW 2
21
Let
5
A=
3
1 2
,
0 4
21
B = 8 6 ,
03
7 4
C = 2 0 ,
9 3
D= 1 9
3 .
Compute each of the following (or, if the expression is undened, say so):
(a) B + C
(b) 4A
(c) AB
(d) BA
(e) BD
(f) DC
Solution:
21
7 4
5 5
(a) B + C = 8 6 + 2 0 = 10 6.
03
9 3
9 6
HW 12
8 17
Show that cfw_x2 1, 2x2 + x is a basis for the subspace S = Spancfw_x 2, x2 + x + 1 of P3 .
Solution:
We check the two properties of basis:
(i) (Spancfw_x2 1, 2x2 + x = S ?) First,
x2 1 = 1(x 2) + 1(x2 + x + 1) and 2x2 + x = 1(x 2) + 2(x2 + x +
HW 6
48
Let S = Spancfw_[2, 4, 1]T .
(a) Give a geometrical description of S .
(b) Find two planes having S as their intersection. (Hint: Use the method for determining
S given in the solution to Example 4.4.1.)
Solution:
(a) S consists of all multiples o
HW 14
10 5
Use a least squares solution to nd a curve of the form y = a cos x + b sin x that best ts
the data points (0, 1), (/2, 2), and (, 0).
Solution: Ideally, the curve would go through all three points, so if we write its equation
as a cos x + b sin
HW 17
13 3
Let C be the vector space of all functions on R having derivatives of all orders. Let
L : C C be the linear function described by multiply by x and then take derivative.
For instance, L(sin x) = (x sin x) = sin x + x cos x (product rule).
(a) I
HW 16
12 3
Find the inverse of the matrix
by using the formula A1 =
Solution:
11
A = 1 2
14
1
3
9
1
Adj A (see Section 12.2).
det A
First,
det A =
1
1
1
11
23
49
= 1(+1)
2
4
3
9
+ 1(1)
= (6) (6) + (2) = 2,
which is nonzero, so the formula can be used.
1
1
HW 15
11 1
Let L : P4 M22 be the linear function given by
L(ax3 + bx2 + cx + d) =
2a b
a
.
d
c + 3d
(a) Show that L is invertible with inverse L1 : M22 P4 given by
L 1
(b) Find p such that L(p) =
2
3
= x3 + ( + 2 )x2 + (3 )x + .
4
.
5
Solution:
3
2
(a) Fo
HW 11
8 10
Let CR denote the set of continuous functions on R (this is a vector space using function
addition and scalar multiplication). Dene L : CR FR by
x
L(f )(x) =
f (t) dt.
0
Show that L is linear.
Solution:
We check the two properties:
(i) For f, g
HW 10
83
Show that FI satises property (V2).
Solution: Let f, g, h FI . We need to show that (f + g ) + h = f + (g + h). Each side
represents a function, so we need to show that (f + g ) + h (x) = f + (g + h) (x) for all
x I . For all x I , we have
(f + g
HW 8
71
Find the dimension of S = Spancfw_x1 , x2 , x3 , where
1
1
1
0
1
1
x1 = , x2 = , x3 = .
1
0
1
1
1
0
Solution: Since the vectors x1 ,
linearly independent. We have
1
1 1 1 1
1 1 0
1 0 1
011
x2 , and x3 span S , they will form a basis if they are
11
HW 7
61
Show that cfw_b1 , b2 is a basis for R2 , where
b1 =
Solution:
2
,
2
b2 =
4
.
2
We check the two properties:
(i) (Spancfw_b1 , b2 = R2 ?) Every linear combination of vectors in R2 is a vector in R2
so Spancfw_b1 , b2 R2 . Let x = [x1 , x2 ]T be
HW 5
41
Let x1 = [1, 3, 2]T and x2 = [4, 7, 1]T .
(a) Determine whether [2, 1, 6]T is in Spancfw_x1 , x2 .
(b) Determine whether [5, 5, 8]T is in Spancfw_x1 , x2 .
Solution:
(a) We are wondering whether there exist numbers 1 and 2 such that [2, 1, 6]T =
1
HW 1
11
In each case, sketch the lines to decide whether the system has a unique solution, no
solution, or innitely many solutions. Then solve the system either by using the methods
of Section 1.1 or by applying row operations to the augmented matrix of t