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Section 2.6 Identifying a Basis (V6)

Definition 2.6.1.

A basis is a linearly independent set that spans a vector space.

The standard basis of \(\IR^n\) is the set \(\{\vec{e}_1, \ldots, \vec{e}_n\}\) where

\begin{align*} \vec{e}_1 &= \left[\begin{array}{c}1 \\ 0 \\ 0 \\ \vdots \\ 0 \\ 0 \end{array}\right] & \vec{e}_2 &= \left[\begin{array}{c}0 \\ 1 \\ 0 \\ \vdots \\ 0 \\ 0 \end{array}\right] & \cdots & & \vec{e}_n = \left[\begin{array}{c}0 \\ 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{array}\right] \end{align*}

For \(\IR^3\text{,}\) these are the vectors \(\vec e_1=\hat\imath=\left[\begin{array}{c}1 \\ 0 \\ 0\end{array}\right], \vec e_2=\hat\jmath=\left[\begin{array}{c}0 \\ 1 \\ 0\end{array}\right],\) and \(\vec e_3=\hat k=\left[\begin{array}{c}0 \\ 0 \\ 1\end{array}\right] \text{.}\)

Observation 2.6.2.

A basis may be thought of as a collection of building blocks for a vector space, since every vector in the space can be expressed as a unique linear combination of basis vectors.

For example, in many calculus courses, vectors in \(\IR^3\) are often expressed in their component form

\begin{equation*} (3,-2,4)=\left[\begin{array}{c}3 \\ -2 \\ 4\end{array}\right] \end{equation*}

or in their standard basic vector form

\begin{equation*} 3\vec e_1-2\vec e_2+4\vec e_3 = 3\hat\imath-2\hat\jmath+4\hat k . \end{equation*}

Since every vector in \(\IR^3\) can be uniquely described as a linear combination of the vectors in \(\setList{\vec e_1,\vec e_2,\vec e_3}\text{,}\) this set is indeed a basis.

Activity 2.6.1.

Label each of the sets \(A,B,C,D,E\) as

  • SPANS \(\IR^4\) or DOES NOT SPAN \(\IR^4\)

  • LINEARLY INDEPENDENT or LINEARLY DEPENDENT

  • BASIS FOR \(\IR^4\) or NOT A BASIS FOR \(\IR^4\)

by finding \(\RREF\) for their corresponding matrices.

\begin{align*} A&=\left\{ \left[\begin{array}{c}1\\0\\0\\0\end{array}\right], \left[\begin{array}{c}0\\1\\0\\0\end{array}\right], \left[\begin{array}{c}0\\0\\1\\0\end{array}\right], \left[\begin{array}{c}0\\0\\0\\1\end{array}\right] \right\} & B&=\left\{ \left[\begin{array}{c}2\\3\\0\\-1\end{array}\right], \left[\begin{array}{c}2\\0\\0\\3\end{array}\right], \left[\begin{array}{c}4\\3\\0\\2\end{array}\right], \left[\begin{array}{c}-3\\0\\1\\3\end{array}\right] \right\}\\ C&=\left\{ \left[\begin{array}{c}2\\3\\0\\-1\end{array}\right], \left[\begin{array}{c}2\\0\\0\\3\end{array}\right], \left[\begin{array}{c}3\\13\\7\\16\end{array}\right], \left[\begin{array}{c}-1\\10\\7\\14\end{array}\right], \left[\begin{array}{c}4\\3\\0\\2\end{array}\right] \right\} & D&=\left\{ \left[\begin{array}{c}2\\3\\0\\-1\end{array}\right], \left[\begin{array}{c}4\\3\\0\\2\end{array}\right], \left[\begin{array}{c}-3\\0\\1\\3\end{array}\right], \left[\begin{array}{c}3\\6\\1\\5\end{array}\right] \right\}\\ E&=\left\{ \left[\begin{array}{c}5\\3\\0\\-1\end{array}\right], \left[\begin{array}{c}-2\\1\\0\\3\end{array}\right], \left[\begin{array}{c}4\\5\\1\\3\end{array}\right] \right\} \end{align*}
Activity 2.6.2.

If \(\{\vec v_1,\vec v_2,\vec v_3,\vec v_4\}\) is a basis for \(\IR^4\text{,}\) that means \(\RREF[\vec v_1\,\vec v_2\,\vec v_3\,\vec v_4]\) doesn't have a non-pivot column, and doesn't have a row of zeros. What is \(\RREF[\vec v_1\,\vec v_2\,\vec v_3\,\vec v_4]\text{?}\)

\begin{equation*} \RREF[\vec v_1\,\vec v_2\,\vec v_3\,\vec v_4] = \left[\begin{array}{cccc} \unknown & \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown & \unknown \\ \end{array}\right] \end{equation*}

Exercises 2.6.1 Exercises

1.

Consider the statement

  1. Write an equivalent statement in terms of other vector properties.
  2. Explain why your statement is true or false.
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{ccc} 3 & 0 & 3 \\ -5 & 3 & -14 \\ -5 & -3 & 4 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & -3 \\ 0 & 0 & 0 \end{array}\right] \end{equation*}
  1. The statement is equivalent to the statement
  2. The set of vectors \(\left\{ \left[\begin{array}{c} 3 \\ -5 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -14 \\ 4 \end{array}\right] \right\} \) is not a basis of \(\mathbb{R}^3\)
2.

Consider the statement

  1. Write an equivalent statement in terms of other vector properties.
  2. Explain why your statement is true or false.
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc} 4 & -5 & -3 & -2 \\ -3 & 1 & 4 & -3 \\ 1 & 4 & -4 & 2 \\ 2 & -4 & -1 & -3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \end{equation*}
  1. The statement is equivalent to the statement
  2. The set of vectors \(\left\{ \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 1 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 2 \\ -3 \end{array}\right] \right\} \) is a basis of \(\mathbb{R}^4\)
3.

Consider the statement

  1. Write an equivalent statement in terms of other vector properties.
  2. Explain why your statement is true or false.
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{ccccc} 2 & -1 & 2 & -1 & 2 \\ -4 & -3 & -14 & -3 & -4 \\ -3 & -1 & -8 & -5 & -3 \\ 1 & 0 & 2 & -2 & 1 \\ 4 & -2 & 4 & -4 & 4 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 2 & 0 & 1 \\ 0 & 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}
  1. The statement is equivalent to the statement
  2. The set of vectors \(\left\{ \left[\begin{array}{c} 2 \\ -4 \\ -3 \\ 1 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ -1 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ -14 \\ -8 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ -5 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ -3 \\ 1 \\ 4 \end{array}\right] \right\} \) is not a basis of \(\mathbb{R}^5\)
4.

Consider the statement

  1. Write an equivalent statement in terms of other vector properties.
  2. Explain why your statement is true or false.
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc} 2 & -2 & -8 & -3 \\ 2 & -4 & -12 & -3 \\ 0 & 4 & 8 & 0 \\ -4 & -1 & 6 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}
  1. The statement is equivalent to the statement
  2. The set of vectors \(\left\{ \left[\begin{array}{c} 2 \\ 2 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} -2 \\ -4 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -8 \\ -12 \\ 8 \\ 6 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ 0 \\ 2 \end{array}\right] \right\} \) is not a basis of \(\mathbb{R}^4\)
5.

Consider the statement

  1. Write an equivalent statement in terms of other vector properties.
  2. Explain why your statement is true or false.
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{ccccc} -1 & -3 & -2 & 3 & -2 \\ -1 & -5 & -5 & 5 & 1 \\ -3 & 0 & -2 & 0 & 1 \\ 3 & 3 & -5 & -3 & 2 \\ 4 & -2 & 1 & 2 & 1 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & -1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}
  1. The statement is equivalent to the statement
  2. The set of vectors \(\left\{ \left[\begin{array}{c} -1 \\ -1 \\ -3 \\ 3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -5 \\ 0 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ -2 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 5 \\ 0 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 1 \\ 2 \\ 1 \end{array}\right] \right\} \) is not a basis of \(\mathbb{R}^5\)

Additional exercises available at checkit.clontz.org.