Answer to Question #5254 in Linear Algebra for Katelynn Bento

Question

Given Matrix A= |123|
|023|
|003|

Find the basis of Col A and the dim of Col A

Answer 1

Notice that the matrix A is upper-triangle, so its determinant is a
product
of diagonal elements:

det(A) = 1*2*3 = 6.

Moreover, det(A) is
non-zero.
Therefore the columns of A are linearly independent and
therefore
they constitutive a basis for R^3, so R^3 = Col(A).

Hence
any base of R^3 fits to be a base for Col(A),
for instance the columns of
A:

(1,0,0), (2,2,0), (3,3,3)

or

(1,0,0), (0,1,0),
(0,0,1)

are the base.

The dimension of Col(A) is the dimension of
R^3 and therefore is equal
to 3.