Linear Algebra, 2020a

Section III. Basis and Dimension 141
3.14 Corollary Where the matrix A is n×n, these statements
(1) the rank of A is n
(2) A is nonsingular
(3) the rows of A form a linearly independent set
(4) the columns of A form a linearly independent set
(5) any linear system whose matrix of coefficients is A has one and only one
solution
are equivalent.
Proof Clearly (1) ⇐⇒ (2) ⇐⇒ (3) ⇐⇒ (4). The last, (4) ⇐⇒ (5), holds
because a set of n column vectors is linearly independent if and only if it is a
basis for R n , but the system
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
a 1,1
a 1,n d 1
a 2,1
c 1 ⎜
⎝
⎟
. ⎠ + ···+ c a 2,n
n ⎜ . ⎝
⎟
. ⎠ = d 2
⎜ . ⎝
⎟
. ⎠
a m,1 a m,n d m
has a unique solution for all choices of d 1 ,...,d n ∈ R if and only if the vectors
of a’s on the left form a basis.
QED
3.15 Remark [Munkres] Sometimes the results of this subsection are mistakenly
remembered to say that the general solution of an m equations, n unknowns
system uses n − m parameters. The number of equations is not the relevant
number; rather, what matters is the number of independent equations, the number
of equations in a maximal independent set. Where there are r independent
equations, the general solution involves n − r parameters.
Exercises
3.16 Transpose each.
⎛ ⎞
( ) ( ) ( ) 0
2 1 2 1 1 4 3
(a)
(b)
(c)
(d) ⎝0⎠
3 1 1 3 6 7 8
0
(e) (−1 −2)
̌ 3.17 Decide if the vector is in the row space of the matrix.
⎛ ⎞
( )
0 1 3
2 1
(a) , (1 0) (b) ⎝−1 0 1⎠, (1 1 1)
3 1
−1 2 7
̌ 3.18 Decide if the vector is in the column space.
⎛
⎞ ⎛ ⎞
( ) ( 1 3 1 1
1 1 1
(a) , (b) ⎝2 0 4 ⎠, ⎝0⎠
1 1 3)
1 −3 −3 0
̌ 3.19 Decide if the vector is in the column space of the matrix.