Linear Algebra

126 Chapter Two. Vector SpacesProof Clearly (1) ⇐⇒ (2) ⇐⇒ (3) ⇐⇒ (4). The last, (4) ⇐⇒ (5), holdsbecause a set of n column vectors is linearly independent if and only if it is abasis for R n , but the system⎛ ⎞ ⎛ ⎞ ⎛ ⎞a 1,1a 1,n d 1a 2,1c 1 ⎜ ⎟⎝ . ⎠ + · · · + c a 2,nn ⎜ ⎟⎝ . ⎠ = d 2⎜ ⎟⎝ . ⎠a m,1 a m,n d mhas a unique solution for all choices of d 1 , . . . , d n ∈ R if and only if the vectorsof a’s form a basis.QEDExercises3.16 Transpose each.( ) 2 1(a)(b)3 1(e) (−1 −2)( ) 2 11 3(c)( 1 4) 36 7 8̌ 3.17 Decide if the vector is in the row space of the matrix.⎛ ⎞( )0 1 32 1(a) , (1 0) (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 11 1 1(a) , (b) ⎝⎠, ⎝ ⎠1 1 3)2 0 41 −3 −300(d)⎛ ⎞0⎝0⎠0̌ 3.19 Decide if the vector is in the column space of the matrix.⎛⎞ ⎛ ⎞( ) ( ) ( ) ( 1 −1 1 22 1 1 4 −8 0(a) , (b) , (c) ⎝⎠, ⎝ ⎠2 5 −3 2 −4 1)̌ 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 −1̌ 3.21 Find ⎛ the rank ⎞of each matrix. ⎛2 1 31 −1⎞2(a) ⎝1 −1 2⎠(b) ⎝ 3 −3 6⎠1 0 3−2 2 −4⎛ ⎞0 0 0(d) ⎝0 0 0⎠0 0 0̌ 3.22 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(c)1 1 −1−1 −1 1⎛1 3⎞2⎝5 1 1⎠6 4 300