Linear Algebra, 2020a

94 Chapter Two. Vector Spaces
(e) Under the inherited operations,
( x
{ ∈ R
y)
2 | x + 3y = 4 and 2x − y = 3 and 6x + 4y = 10}
1.23 Define addition and scalar multiplication operations to make the complex
numbers a vector space over R.
1.24 Is the set of rational numbers a vector space over R under the usual addition
and scalar multiplication operations?
1.25 Show that the set of linear combinations of the variables x, y, z is a vector space
under the natural addition and scalar multiplication operations.
1.26 Prove that this is not a vector space: the set of two-tall column vectors with
real entries subject to these operations.
( ) ( ) ( )
x1 x2 x1 − x 2
+ =
y 1 y 2 y 1 − y 2
( x
r · =
y)
( ) rx
ry
1.27 Prove or disprove that R 3 is a vector space under these operations.
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
x 1 x 2 0
x rx
(a) ⎝y 1
⎠ + ⎝y 2
⎠ = ⎝0⎠ and r ⎝y⎠ = ⎝ry⎠
z 1 z 2 0
z rz
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
x 1 x 2 0
x 0
(b) ⎝y 1
⎠ + ⎝y 2
⎠ = ⎝0⎠ and r ⎝y⎠ = ⎝0⎠
z 1 z 2 0
z 0
̌ 1.28 For each, decide if it is a vector space; the intended operations are the natural
ones.
(a) The diagonal 2×2 matrices
( ) a 0
{ | a, b ∈ R}
0 b
(b) This set of 2×2 matrices
( )
x x+ y
{
| x, y ∈ R}
x + y y
(c) This set
⎛ ⎞
x
{ ⎜y
⎟
⎝ z ⎠ ∈ R4 | x + y + z + w = 1}
w
(d) The set of functions {f: R → R | df/dx + 2f = 0}
(e) The set of functions {f: R → R | df/dx + 2f = 1}
̌ 1.29 Prove or disprove that this is a vector space: the real-valued functions f of one
real variable such that f(7) =0.
̌ 1.30 Show that the set R + of positive reals is a vector space when we interpret ‘x + y’
to mean the product of x and y (so that 2 + 3 is 6), and we interpret ‘r · x’ asthe
r-th power of x.
1.31 Is {(x, y) | x, y ∈ R} a vector space under these operations?
(a) (x 1 ,y 1 )+(x 2 ,y 2 )=(x 1 + x 2 ,y 1 + y 2 ) and r · (x, y) =(rx, y)
(b) (x 1 ,y 1 )+(x 2 ,y 2 )=(x 1 + x 2 ,y 1 + y 2 ) and r · (x, y) =(rx, 0)