# Answer to Question #5254 in Linear Algebra for Katelynn Bento

Question #5254

Given Matrix A= |123|

|023|

|003|

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

|023|

|003|

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

Expert's answer

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.

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.

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