Question

*Find a basis for the subspace C(subset)R3 spanned by the columns of A.
*Find a basis for the subspace R(subset)R4 spanned by the rows of A.

Answer 1

Now the subspace C, are just the two vectors of the original matrix with pivots?

And the subspace R, are just the vectors with the free variables in reduced echelon form?

Answer 2

\[A=\left[\begin{matrix}1 & 0 & \frac{ 16 }{ 5 } & \frac{ 17 }{ 5 }\\ 0 & 1 & \frac{ -7 }{ 5 } & \frac{ 1 }{ 5 }\\ 0 & 0 & 0 & 0\end{matrix}\right]\]

this is the reduced form of the original matrix.

*so the basis for the subspace C will be the first two columns of my original matrix?
*and the basis for subspace R will be the vectors with the free variables x3 and x4 right?

Answer 3

and the dimC and dimR = 2, are they always going to be the same?

Answer 4

for the first part u r right. the basis for C will be the columns of the original matrix that have pivots in the row-reduced matrix.

Answer 5

for the second part, a basis for R will be the remaining rows after row-reduction.

Answer 6

so the basis for R is just those first two rows in the matrix A above?
then is the dimR=4?

Answer 7

yes and no. dim R=2

Answer 8

u have only two non-zero rows there.

Answer 9

so if i am looking for the dimR i count the rows, and if im looking for the dimC i count the columns? will they always be the same number?

Answer 10

u count the remaining rows for the row-space and the columns WITH pivots for the column-space.

Answer 11

thank you. just one more quick post if you have time.

Answer 12

sure.