Let A and B be row equivalent matrices. (a) Show that the dimension of the column space of A equals the dimension of the column space of B. (b) Are the column spaces of the two matrices necessarily the same? Justify your answer.
Let A and B be row equivalent matrices.
(a) Show that the dimension of the column space
of A equals the dimension of the column space
of B.
(b) Are the column spaces of the two matrices
necessarily the same? Justify your answer.
Here, A and B are two row equivalent matrices.
a) The dimension of the column space of A equals the dimension of the column space of B has to be shown.
b)The column spaces of the two matrices are necessarily the same has to be justified.
As A and B are the row equivalent matrices, the matrices must have same row-spaces. It means the matrices have same row-echelon form.
This echelon form is obtained by applying same row -operations to the both matrices.
The matrices A and B are transformed to to its row -echelon forms. The matrices's column spaces must be preserved.
A matrix's column space is subspace and it is spanned by the columns.
Hence, the dimension of the column space of A equals the dimension of the column space of B.
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