In Exercises 19 and 20, \(V\) is a vector space. Mark each statement True or False. Justify each answer.

19.

a. The number of pivot columns of a matrix equals the dimension of its column space.

b. A plane in \({\mathbb{R}^3}\) is a two-dimensional subspace of \({\mathbb{R}^3}\).

c. The dimension of the vector space \({{\mathop{\rm P}\nolimits} _4}\) is 4.

d. If \(\dim V = n\) and \(S\) is a linearly independent set in \(V\), then \(S\) is a basis for \(V\).

e. If a set \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) spans a finite-dimensional vector space \(V\) and if \(T\) is a set of more than p vectors in \(V\), then \(T\) is linearly dependent.

a)

Thedimension of \({\mathop{\rm Nul}\nolimits} A\) is thenumber of free variablesin the equation \(A{\mathop{\rm x}\nolimits} = 0\), and the dimension of \({\mathop{\rm Col}\nolimits} A\)is thenumber of pivot columnsin \(A\).

Thus, the given statement (a) is true.

b)

A plane in \({\mathbb{R}^3}\) is athree-dimensional subspace of \({\mathbb{R}^3}\).

Thus, the given statement (b) is false.

c)

Thestandard basisfor \({\mathbb{R}^n}\) contains \(n\) vectors; so \(\dim {\mathbb{R}^n} = n\). The standard polynomial basis \(\left\{ {1,t,{t^2}} \right\}\) shows that \({{\mathop{\rm P}\nolimits} _2} = 3\). In general, \(\dim {{\mathop{\rm P}\nolimits} _n} = n + 1\).

Thus, the given statement (c) is false.

d)

Theorem 12states that let \(V\) be a p-dimensional vector space; \(p \ge 1\), then any linearly independent set of exactly \(p\) elements in \(V\)is automatically a basis for \(V\). Any set of exactly \(p\) elements that span \(V\)is automatically a basis for \(V\).

The set \(S\) must have \(n\) elements.

Thus, the given statement (d) is false.

e)

Theorem 9states that if a vector space \(V\) has abasis \(B = \left\{ {{{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}} \right\}\), then any set in \(V\)containing more than \(n\) vectors must belinearly dependent.

Thus, the given statement (e) is true.