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Every product $A\vec{x}$ is a member of the column space $C(A)$ because matrix-vector multiplication is defined as a linear combination of the columns of $A$. Specifically, if $A = [\vec{a}_1 \ \vec{a}_2 \ \cdots \ \vec{a}_k]$ and $\vec{x} = [x_1, \dots, x_k]^T$, then $A\vec{x} = x_1\vec{a}_1 + \dots + x_k\vec{a}_k$.

Conditions: $A$ is an $n \times k$ matrix; $\vec{x} \in \mathbb{R}^k$; $C(A)$ is the span of the columns of $A$

Every product $A\vec{x}$ is a member of the column space $C(A)$ because matrix-vector multiplication is defined as a linear combination of the columns of $A$. Specifically, if $A = [\vec{a}_1 \ \vec{a}_2 \ \cdots \ \vec{a}_k]$ and $\vec{x} = [x_1, \dots, x_k]^T$, then $A\vec{x} = x_1\vec{a}_1 + \dots + x_k\vec{a}_k$.

Conditions: $A$ is an $n \times k$ matrix; $\vec{x} \in \mathbb{R}^k$; $C(A)$ is the span of the columns of $A$