Proving that vector is in row space of Matrix

Question

(a) If $A$ is an m×n matrix, show that $v \in Row(A)$ if and only if $v = x^tA$ for some $x ∈R^m$.

(b) Using the above result, show that $Row(A + B) \subseteq Row(A) + Row(B)$.

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Part A.

I"m completely stuck here. All I've been able to do is go round in circles.

First, I assumed that, if $v = x^tA$, then $x^tA \in Col(A)$. Transposing both sides; $A^tx \in Row(A)$. That's about as far as I've gotten.

Am I taking this in the right direction? If so, where can I go from here?

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I'm not even sure if I can do Part B without Part A, but here goes;

Let $Row(A) = U$, $Row(B) = V$, where $U$ and $V$ are subspaces of $R^n$. Let $u, v$ be vectors in U and V respectively.

In this case, am I right in thinking that $Row(A + B) \subseteq Row(A) + Row(B)$ seems quite similar to asking whether $(u + v) \in U + V$?

Answer 1

Let $A = \begin{bmatrix} r_{A1} \\ \vdots \\ r_{Am} \end{bmatrix}$ and $B = \begin{bmatrix} r_{B1} \\ \vdots \\ r_{Bm} \end{bmatrix}$.

part a):

$v \in row(A)$ if and only if

$\exists x_i$ such that $$v=\sum_{i=1}^mx_ir_{Ai}$$

which is equivalent to $$v=\begin{bmatrix} x_1 & \ldots & x_m\end{bmatrix}\begin{bmatrix} r_{A1} \\ \vdots \\ r_{Am} \end{bmatrix} = x^TA.$$

part b)

If $v \in row(A+B)$, then $\exists x$ such that $v=x^T(A+B)=x^TA+x^TB$.

since $x^TA \in row(A)$ and $x^TB \in row(B)$.

$$row(A+B) \subseteq row(A) + row(B).$$