QuestionSuppose a $7 \times 11$ matrix $A$ has seven pivot columns. Is $\operatorname{Col} A=\mathbb{R}^{7} ?$ Is $N u l=\mathbb{R}^{4}$ ? Explain your answers.
Is $\operatorname{Col} A=\mathbb{R}^{7}$ ? A: No, $\operatorname{Col} A$ is not $\mathbb{R}^{7}$. Since $A$ has seven pivot columns, $\operatorname{dim} \operatorname{Col} A=4$. Thus, $\operatorname{Col} A$ is equal to $\mathbb{R}^{4}$. B. No. Since $A$ has seven pivot columns, $\operatorname{dim} \operatorname{Col} A=7$. Thus, $\operatorname{Col} A$ is a seven-dimensional subspace of $\mathbb{R}^{7}$, so $\mathrm{Col} A$ is not equal to $\mathbb{R}^{7}$. C. Yes. Since $A$ has seven pivot columns, $\operatorname{dim} \operatorname{Col} A=7$. Thus, $\operatorname{Col} A$ is a seven-dimensional subspace of $\mathbb{R}^{7}$, so $\operatorname{Col} A$ is equal to $\mathbb{R}^{7}$. D. No, the column space of $A$ is not $\mathbb{R}^{7}$. Since $A$ has seven pivot columns, $\operatorname{dim} \operatorname{Col} A=0$. Thus, $\operatorname{Col} A$ is equal to 0 .
Is $\operatorname{Nul} A=\mathbb{R}^{4}$ ? A. No, Nul $A$ is not equal to $\mathbb{R}^{4}$. It is true that $\operatorname{dim} \operatorname{Nul} A=4$, but Nul $A$ is a subspace of $\mathbb{R}^{11}$. B. No, Nul $A$ is not equal to $\mathbb{R}^{4}$. Since $A$ has seven pivot columns, $\operatorname{dim} \operatorname{Nul} A=7$. Thus, Nul $A$ is equal to $\mathbb{R}^{7}$. C. No, Nul $A$ is equal to $\mathbb{R}^{4}$. Since $A$ has seven pivot columns, $\operatorname{dim} \operatorname{Nul} A=0$. Thus, Nul $A$ is equal to 0 . D. Yes, Nul $A$ is equal to $\mathbb{R}^{4}$. Since $A$ has seven pivot columns, $\operatorname{dim} \operatorname{Nul} A=4$. Thus, Nul $A$ is equal to $\mathbb{R}^{4}$.