Linear Algebra

Section III. Basis and Dimension 129Exercises3.16 Transpose each.( )2 1(a)(b)3 1(e) ( −1 −2 )( )2 11 3(c)( )1 4 36 7 8(d)̌ 3.17 Decide if the vector is in the row space of the matrix.( )2 1(a) , ( 1 0 ) ( )0 1 3(b) −1 0 1 , ( 1 1 1 )3 1−1 2 7̌ 3.18 Decide if the vector is in the column space.( ) ( ( ) ( )1 3 1 1 1 1 1(a) , (b) 2 0 4 , 01 1 3)1 −3 −3 0̌ 3.19 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 −1̌ 3.20 Find ( the rank)of each matrix.2 1 3(a) 1 −1 2 (b)1 0 3(d)( 0 0 00 0 00 0 0)( 1 −1 23 −3 6−2 2 −4)(c)( ) 000( 1 3) 25 1 16 4 3̌ 3.21 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 31 −1 −3(c) {1( + x, 1 − x 2 ), 3 + ( 2x − x 2 })⊆( P 3)1 0 1 1 0 3 −1 0 −5(d) {,,} ⊆ M3 1 −1 2 1 4 −1 −1 −92×33.22 Which matrices have rank zero? Rank one?̌ 3.23 Given a, b, c ∈ R, what choice of d will cause this matrix to have the rank ofone? ( )a bc d3.24 Find the column rank of this matrix.( )1 3 −1 5 0 42 0 1 0 4 13.25 Show that a linear system with at least one solution has at most one solutionif and only if the matrix of coefficients has rank equal to the number of its columns.̌ 3.26 If a matrix is 5×9, which set must be dependent, its set of rows or its set ofcolumns?