Question

In Exercises 15 and 16, mark each statement True or False. Justify each answer. Unless stated otherwise, \(B\) is a basis for a vector space \(V\).

In Exercises 15 and 16, mark each statement True or False. Justify each answer. Unless stated otherwise, \(B\) is a basis for a vector space \(V\).

16.

1. If\(B\)is the standard basis for\({\mathbb{R}^n}\)then the\(B\)-coordinate vector of an\({\mathop{\rm x}\nolimits} \)in\({\mathbb{R}^n}\)is x itself.
2. The correspondence\({\left( {\mathop{\rm x}\nolimits} \right)_B} \mapsto {\mathop{\rm x}\nolimits} \)is called coordinate mapping.
3. In some cases, a plane in\({\mathbb{R}^3}\)can be isomorphic to\({\mathbb{R}^2}\).

Short answer

a. The given statement is true.

b. The given statement is false.

c. The given statement is true.

Step-by-step solution

Determine whether the given statement is true or false

a)

The entries in vector\({\mathop{\rm x}\nolimits} = \left( {\begin{array}{*{20}{c}}1\\6\end{array}} \right)\)are thecoordinates of\({\mathop{\rm x}\nolimits} \)relative to the standard basis \(\varepsilon = \left\{ {{{\mathop{\rm e}\nolimits} _1},{{\mathop{\rm e}\nolimits} _2}} \right\}\). If\(\varepsilon = \left\{ {{{\mathop{\rm e}\nolimits} _1},{{\mathop{\rm e}\nolimits} _2}} \right\}\), then\({\left( {\mathop{\rm x}\nolimits} \right)_\varepsilon } = {\mathop{\rm x}\nolimits} \).

Thus, statement (a) is true.

Determine whether the given statement is true or false

b)

Let\(B = \left\{ {{{\mathop{\rm b}\nolimits} _1},{{\mathop{\rm b}\nolimits} _2},...,{{\mathop{\rm b}\nolimits} _n}} \right\}\)be a basis forvector space\(V\). Then the coordinate mapping \({\mathop{\rm x}\nolimits} \mapsto {\left( {\mathop{\rm x}\nolimits} \right)_B}\)is one-to-one linear transformation from\(V\)onto\({\mathbb{R}^n}\).

Thus, statement (b) is false.

Determine whether the given statement is true or false

c)

The plane isisomorphic to\({\mathbb{R}^2}\)if it passes through theorigin, as shown in Example 7.

Thus, statement (c) is true.