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In Exercises 17 and $18,$ mark each statement True or False. Justify each answer. Here $A$ is an $m \times n$ matrix.a. If $\mathcal{B}=\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}\right\}$ is a basis for a subspace $H$ and if $\mathbf{x}=c_{1} \mathbf{v}_{1}+\cdots+c_{p} \mathbf{v}_{p},$ then $c_{1}, \ldots, c_{p}$ are the coordinates of $\mathbf{x}$ relative to the basis $\mathcal{B}$ .b. Each line in $\mathbb{R}^{n}$ is a one-dimensional subspace of $\mathbb{R}^{n}$ .c. The dimension of Col $A$ is the number of pivot columns of $A .$d. The dimensions of Col $A$ and Nul $A$ add up to the number of columns of $A$ .e. If a set of $p$ vectors spans a $p$ -dimensional subspace $H$ of $\mathbb{R}^{n},$ then these vectors form a basis for $H .$

A. TRUEB. FALSEC. TRUED. TRUEE. TRUE

Algebra

Chapter 2

Matrix Algebra

Section 9

Dimension and Rank

Introduction to Matrices

Missouri State University

Baylor University

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Okay, so this question will be answering with a serious off a tree and forces. So it was a if b is a basis for subspace hated in each. Better in hate can be ridden in only one way as linear combination of factors in B. This fact is true, which is, quite literally, the definition of a basis. And if you need more help on this question, if you go to the text book page 1 55 the beginning off section to 2.9 on the second paragraph, it says, is basically verbatim. Question be if B is a basis for suspicion of hate in the correspondence, makes hatred look and act the same as our p. So this is also true, which is literally told bourbon in page 1 57 See the dimensions off mo off a number of variables in the equation X because is there this is false is equal to the number off free variables in a executed the dimension off The column Space off A is rank A. This is true because this is that that this is the definition off, Frank. Hey, if head is a p dimensional subspace, there linearly independent set off the vectors in hedge is a basis for Paige. This is true by the base system.

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