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In Exercises 31–36, respond as comprehensively as possible, and justify your answer.If $P$ is a $5 \times 5$ matrix and Nul $P$ is the zero subspace, what can you say about solutions of equations of the form $P \mathbf{x}=\mathbf{b}$ for $\mathbf{b}$ in $\mathbb{R}^{5} ?$

If $\mathrm{NulP}=\{0\},$ then the equation $P x=0$ has only the trivial solution. Since $\mathrm{P}$ is square, the IMT shows that $P$ is invertible and the equation $P x=b$ has a solution for each $\mathrm{bin} R^{5} .$ 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

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So for this question, say's the no space off B is the zero subspace That is, of course, to zero. Then this implies that speedy multiply by extra quarter zero as only the trivial solution as only a trivial solution. What I mean by this is dots only X equal to zero. That's what I mean by tree Vo con Mick P. Times X equals zero. No, says p squared. Then we can use the invincible metrics Xyrem. Not since section Superman 30 and we have That's hey is invincible. And what this implies is that P eggs equal Toby as unique solution for every be in our five.

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