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Chapter 1: Q34E (page 1)

Each statement in Exercises 33-38 is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If a statement is true, give a justification. (One specific example cannot explain why a statement is always true. You will have to do more work here than in Exercises 21 and 22.)

34. If \({{\mathop{\rm v}\nolimits} _1},...,{v_4}\) are in \({\mathbb{R}^4}\) and \({{\mathop{\rm v}\nolimits} _3} = 0\), then \(\left\{ {{v_1},{v_2},{v_3},{v_4}} \right\}\) is linearly dependent.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Determine whether the given statement is true or false

If set \(S = \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{v_p}} \right\}\) in \({\mathbb{R}^n}\) contains the zero vector, then it islinearly dependent.

Thus, the given statement is true.

02

Determine why the given statement is true

Since the set \(\left\{ {{v_1},{v_2},{v_3},{v_4}} \right\}\) in \({\mathbb{R}^4}\) contains the zero vector, \({{\mathop{\rm v}\nolimits} _3} = 0\), the vectors in \({\mathbb{R}^4}\) are linearly dependent.

Thus, the given statement is true.

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Most popular questions from this chapter

Question: If A is a non-zero matrix of the form,[a-bba] then the rank of A must be 2.

Write the vector \(\left( {\begin{array}{*{20}{c}}5\\6\end{array}} \right)\) as the sum of two vectors, one on the line \(\left\{ {\left( {x,y} \right):y = {\bf{2}}x} \right\}\) and one on the line \(\left\{ {\left( {x,y} \right):y = x/{\bf{2}}} \right\}\).

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