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In Exercises 31–36, respond as comprehensively as possible, and justify your answer.If $Q$ is a $4 \times 4$ matrix and $\mathrm{Col} Q=\mathbb{R}^{4},$ what can you say about solutions of equations of the form $Q \mathbf{x}=\mathbf{b}$ for $\mathbf{b}$ in $\mathbb{R}^{4} ?$

If $C o l Q=R^{4},$ then the columns of $Q \operatorname{span} R^{4} .$ since $Q$ is square, the IMT shows that $Q$ is invertible and the equation $Q x=b$ has a solution for each $\mathrm{b}$ in $R^{4} .$ Also, each solution is unique, by Theorem 5 in Section $2.2 .$

Algebra

Chapter 2

Matrix Algebra

Section 8

Subspaces of Rn

Introduction to Matrices

Oregon State University

McMaster University

Harvey Mudd College

Lectures

01:32

In mathematics, the absolu…

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In Exercises 31–36, respon…

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Suppose $A$ is a $4 \times…

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Using matrix algebra techn…

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Solve each system using ma…

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Suppose that $A$ and $B$ a…

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Let $A$ and $B$ be $4 \tim…

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So we have that Q is a Square four by four matrix and that the column space of Q is equal to form. But there are only four columns of Q. So this means that the columns of Q are linearly independent. Therefore, a Q is in vertebral by the in vertebral matrix there. Um, moreover, Q X equals B has the unique solution X equals the inverse of Q times. Be, um, for each B and R four, and that's by three and five of section 2.2, and that completes the problem.

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