Linear Algebra, 2020a

142 Chapter Two. Vector Spaces
(a)
( ) ( 2 1 1
,
2 5 −3)
(b)
( ) ( 4 −8 0
,
2 −4 1)
(c)
⎛
⎞ ⎛ ⎞
1 −1 1 2
⎝ 1 1 −1⎠,
⎝0⎠
−1 −1 1 0
̌ 3.20 Find a basis for the row space of this matrix.
⎛
2 0 3
⎞
4
⎜0 1 1 −1
⎟
⎝3 1 0 2 ⎠
1 0 −4 −1
̌ 3.21 Find ⎛ the rank ⎞of each matrix. ⎛
2 1 3
1 −1
⎞
2
⎛
1 3
⎞
2
(a) ⎝1 −1 2⎠
(b) ⎝ 3 −3 6 ⎠ (c) ⎝5 1 1⎠
1 0 3
−2 2 −4
6 4 3
⎛ ⎞
0 0 0
(d) ⎝0 0 0⎠
0 0 0
3.22 Give a basis for the column space of this matrix. Give the matrix’s rank.
⎛
1 3 −1
⎞
2
⎝2 1 1 0⎠
0 1 1 4
̌ 3.23 Find a basis for the span of each set.
(a) {(1 3), (−1 3), (1 4), (2 1)} ⊆ M 1×2
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 3 1
(b) { ⎝2⎠ , ⎝ 1 ⎠ , ⎝−3⎠} ⊆ R 3
1 −1 −3
(c) {1 + x, 1 − x 2 ,3+ 2x − x 2 } ⊆ P 3
(d) {
( 1 0 1
3 1 −1
)
,
( 1 0 3
2 1 4
)
,
( −1 0 −5
−1 −1 −9
)
} ⊆ M 2×3
3.24 Give a basis for the span of each set, in the natural vector space.
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 −1 0
(a) { ⎝1⎠ , ⎝ 2 ⎠ , ⎝12⎠}
3 0 6
(b) {x + x 2 ,2− 2x,7,4+ 3x + 2x 2 }
3.25 Which matrices have rank zero? Rank one?
̌ 3.26 Given a, b, c ∈ R, what choice of d will cause this matrix to have the rank of
one? ( a
) b
c d
3.27 Find the column rank of this matrix.
( )
1 3 −1 5 0 4
2 0 1 0 4 1
3.28 Show that a linear system with at least one solution has at most one solution if
and only if the matrix of coefficients has rank equal to the number of its columns.
̌ 3.29 If a matrix is 5×9, which set must be dependent, its set of rows or its set of
columns?