In Exercises 17 and 18, mark each statement True or False. Justify each answer. Here A is an \(m \times n\) matrix.

a. If \(B = \left\{ {{{\bf{v}}_{\bf{1}}},\,...,\,{{\bf{v}}_p}} \right\}\) is a basis for a subspace H and if \({\bf{x}} = {c_{\bf{1}}}{{\bf{v}}_{\bf{1}}} + .... + {c_p}{{\bf{v}}_p}\), then \({c_{\bf{1}}}\),…., \({c_p}\)are the coordinates of x relative to the basis B.

b. Each line in \({\mathbb{R}^n}\) is an 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 up p vectors spans a p-dimensional subspace H of \({\mathbb{R}^n}\), then these vectors form a basis for H.

Step by step solution

01

Check for statement (a)

As per the basis theorem, the given statement is true.

02

Check for statement (b)

One dimensional subspace of \({\mathbb{R}^n}\) is in the span of v. The span ofv represents a line on the 2D plane. The line that does not pass through the origin is not a subspace of \({\mathbb{R}^n}\).

03

Check for statement (c)

Thepivot columns of A are linearly independent. So, they form a basis of Col A.

The number of pivot columns of A is equal to the dimensions of Col A.

04

Check for statement (d)

By the rank theorem, \[{\rm{rank}}\,A + \dim \,{\rm{Nul}}\,A = n\].

So, the given statement is true.

05

Check for statement (e)

By the basis theorem,if p vectors spans a p-dimensional subspace H of \({\mathbb{R}^n}\), then these vectors form a basis for H.

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