There's a question in charles curtis linear algebra book which states:(adsbygoogle = window.adsbygoogle || []).push({});

Let ##f1, f2, f3## be functions in ##\mathscr{F}(R)##.

a. For a set of real numbers ##x_{1},x_{2},x_{3}##, let ##(f_{i}(x_{j}))## be the ##3-by-3## matrix

whose (i,j) entry is ##(f_{i}(x_{j}))##, for ##1\leq i,j \leq 3##. Prove that ##f_{1}, f_{2}, f_{3}## are linearly independent if the rows of the matrix ##(f_{i}(x_{j}))## are linearly independent.

Obviously if ##f_1,f_2,f_3## are in terms of basis vectors than they are linearly independent.

But can I say that if matrix ##A = (f_{i}(x_{j}))## is linearly independent, then they are in echelon form.

Therefore, I can row reduce the matrix to a diagonal matrix s.t. ## a_{i,i} \neq 0##.

Since the rows are linearly independent then ##f_{i} \neq 0 \quad 1 \leq i \leq 3##, therefore for

##\alpha_{i} f_{i} = 0## only if ##\alpha_{i} = 0##.

Is this a good proof for that question?

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Show linear independence

Loading...

Similar Threads - Show linear independence | Date |
---|---|

I For groups, showing that a subset is closed under operation | Feb 20, 2017 |

Show that a set of functions is linearly independent | Nov 10, 2012 |

Show that linear transformation is surjective but not injective | Feb 5, 2012 |

Linear algebra show eigenvalue | Mar 2, 2008 |

Linear transformation, show surjection and ker=0. | Apr 13, 2005 |

**Physics Forums - The Fusion of Science and Community**