Question

hi quick linear algebra question. please help

I understand that if the determinant of a coefficient matrix is non-zero then the set spans something else.

I would like to know if the determinant of the coefficient matrix is 0 then does that mean that the set of vectors does not span the space?

Answer 1

it spans still, just not be the basis

Answer 2

is it not a basis because it would be linearly dependent? @OOOPS

Answer 3

span \(\neq \) basis, you know it?
span + linearly independent --> basis
and if basis--> span (no backward)

Answer 4

yeah if either it does not span or it is linearly dependent then no basis right?

Answer 5

for example \[\left[\begin{matrix}1&1\\2&0\\\end{matrix}\right]\] , they are basis of R^2 --> they span R^2

Answer 6

\[\left[\begin{matrix}1&-1\\1&-1\end{matrix}\right]\] they span R^2 too, but they are not the basis since their cols are not linearly independent, det = 0

Answer 7

ok that makes sense. thanks.

Answer 8

ok