Linear Algebra

Section III. Basis and Dimension 1233.7 Example Given this matrix,⎛ ⎞1 3 72 3 8⎜ ⎟⎝0 1 2⎠4 0 4to get a basis for the column space, temporarily turn the columns into rows andreduce.⎛⎜1 2 0 4⎞⎛⎟⎝3 3 1 0⎠ −3ρ 1+ρ 2 −2ρ 2 +ρ 3 ⎜1 2 0 4⎞⎟−→ −→ ⎝0 −3 1 −12⎠−7ρ 1 +ρ 37 8 2 40 0 0 0Now turn the rows back to columns.⎛ ⎞ ⎛ ⎞1 02〈 ⎜ ⎟⎝0⎠ , −3⎜ ⎟⎝ 1⎠ 〉4 −12The result is a basis for the column space of the given matrix.3.8 Definition The transpose of a matrix is the result of interchanging the rowsand columns of that matrix, so that column j of the matrix A is row j of A trans ,and vice versa.So we can summarize the prior example as “transpose, reduce, and transposeback.”We can even, at the price of tolerating the as-yet-vague idea of vector spacesbeing “the same,” use Gauss’s Method to find bases for spans in other types ofvector spaces.3.9 Example To get a basis for the span of {x 2 + x 4 , 2x 2 + 3x 4 , −x 2 − 3x 4 } inthe space P 4 , think of these three polynomials as “the same” as the row vectors(0 0 1 0 1), (0 0 2 0 3), and (0 0 −1 0 −3), apply Gauss’sMethod⎛ ⎞0 0 1 0 1⎜ ⎟⎝ 0 0 2 0 30 0 −1 0 −3−→−→⎠ −2ρ 1+ρ 2 2ρ 2 +ρ 3ρ 1 +ρ 3⎛⎞0 0 1 0 1⎜⎟⎝0 0 0 0 1⎠0 0 0 0 0and translate back to get the basis 〈x 2 + x 4 , x 4 〉. (As mentioned earlier, we willmake the phrase “the same” precise at the start of the next chapter.)Thus, our first point in this subsection is that the tools of this chapter giveus a more conceptual understanding of Gaussian reduction.For the second point of this subsection, observe that row operations on amatrix can change its column space.(1 22 4)( )1 2−→0 0−2ρ 1 +ρ 2