Question

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

20.

a. \({\mathbb{R}^2}\) is a two-dimensional subspace of \({\mathbb{R}^3}\).

b. The number of variables in the equation \(A{\mathop{\rm x}\nolimits} = 0\) equals the dimension of \({\mathop{\rm Nul}\nolimits} A\).

c. A vector space is infinite-dimensional if it is spanned by an infinite set.

d. If \(\dim V = n\) and if \(S\) spans \(V\)then \(S\) is a basis for \(V\).

e. The only three-dimensional subspace of \({\mathbb{R}^3}\) is \({\mathbb{R}^3}\) itself.

Short answer

1. The given statement is false.
2. The given statement is false.
3. The given statement is false.
4. The given statement is false.
5. The given statement is true.

Step-by-step solution

Determine whether the given statement is true or false

a)

There is no subset of set \({\mathbb{R}^2}\) within \({\mathbb{R}^3}\).

Thus, the given statement (a) is false.

Determine whether the given statement is true or false

b)

Thedimensionof \({\mathop{\rm Nul}\nolimits} A\) is the number of free variablesin the equation \(A{\mathop{\rm x}\nolimits} = 0\).

Thus, the given statement (b) is false.

Determine whether the given statement is true or false

c)

A basis could still contain a finite number of elements, which would make the vector space finite-dimensional.

Thus, the given statement (c) is false.

Determine whether the given statement is true or false

d)

Theorem 12states that let \(V\) be a p-dimensional vector space; \(p \ge 1\), then anylinearly independent setof 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.

Determine whether the given statement is true or false

e)

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

Thus, the given statement (e) is true.