proving row space column space

**transgalactic** · June 21st 2011, 03:12 PM

A , B are nXn matrices
and
AB=(A)^t
t-is transpose
prove that the space spanned by A's row equals the space spanned by A's columns
i know that there dimentions are equals
so in order to prove equality i need to prove that one is a part of the other
how to do it?

**NonCommAlg** · June 21st 2011, 03:25 PM

Originally Posted by transgalactic

A , B are nXn matrices
and
AB=(A)^t
t-is transpose
prove that the space spanned by A's row equals the space spanned by A's columns
i know that there dimentions are equals
so in order to prove equality i need to prove that one is a part of the other
how to do it?

an element in the row space of $A$ is in the form $A^tx$ , where $x$ in a vector. an element in the column space of $A$ is in the form $Ay$ , where $y$ is a vector. so if the vector $x$ is given and we let $Bx = y,$ then $A^tx = Ay$ . this proves that the row space of $A$ is a subspace of the column space of $A$ .

**transgalactic** · June 21st 2011, 04:04 PM

each column i of (AB)_i=A*B_i
i was told by my proff that that column i of AB is a member from the span of the columns of A

but i dont get this result
suppose the member of B_i column is (c1,c2,..,cn)
so the multiplication of A by the B_i column
we get then the first member is dot product from the first row with (c1,c2,..,cn)
i cant see how its a variation from the A columns?

**NonCommAlg** · June 21st 2011, 04:09 PM

the key is this simple fact that if $x$ is a vector with entries $x_1, \ldots, x_n$ and $v_1, \ldots v_n$ are the columns of an $n \times n$ matrix $C$ , then $Cx = x_1v_1 + \ldots + x_n v_n$ , which is an element of the column space of $C$ . it should be clear now that $C^tx$ is an element of the row space of $C$ .

**transgalactic** · June 21st 2011, 04:24 PM

i cant get the diagram
of a coefficient next to each of A's columns from this coulmn by matrix multiplication

**NonCommAlg** · June 21st 2011, 04:32 PM

Originally Posted by transgalactic

i cant get the diagram
of a coefficient next to each of A's columns from this coulmn by matrix multiplication

well, i suggest you take $2 \times 2$ matrices first to get an idea.
let $C = \begin{pmatrix}a & b \\ c & d \end{pmatrix}.$ so the columns are $v_1=\begin{pmatrix}a \\ c \end{pmatrix}$ and $v_2 = \begin{pmatrix}b \\ d \end{pmatrix}$ . now let $x = \begin{pmatrix}x_1 \\ x_2 \end{pmatrix}$ . show that $Cx=x_1v_1 + x_2v_2.$

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