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Chapter 3: Problem 12

Which of the sets that follow are spanning sets for \(\mathbb{R}^{3} ?\) Justify your answers. (a) \(\left\\{(1,0,0)^{T},(0,1,1)^{T},(1,0,1)^{T}\right\\}\) (b) \(\left\\{(1,0,0)^{T},(0,1,1)^{T},(1,0,1)^{T},(1,2,3)^{T}\right\\}\) (c) \(\left\\{(2,1,-2)^{T},(3,2,-2)^{T},(2,2,0)^{T}\right\\}\) (d) \(\left\\{(2,1,-2)^{T},(-2,-1,2)^{T},(4,2,-4)^{T}\right\\}\) (e) \(\left\\{(1,1,3)^{T},(0,2,1)^{T}\right\\}\)

Short Answer

Expert verified
Only set (a) is a spanning set for \(\mathbb{R}^{3}\), as its vectors are linearly independent and form a basis for \(\mathbb{R}^{3}\). Sets (b), (c), (d), and (e) are not spanning sets for \(\mathbb{R}^{3}\) due to either having linearly dependent vectors or an insufficient number of vectors.

Step by step solution

01

Set (a)

The set (a) consists of the vectors \((1,0,0)^{T},(0,1,1)^{T},(1,0,1)^{T}\). We can create a matrix with these vectors as columns: \[A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \\ \end{pmatrix} \] We need to find if the vectors are linearly independent. To do this, we perform Gaussian elimination to find the Reduced Row Echelon Form (RREF) of A: \[RREF(A) = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} \] Since the RREF of A has 3 pivot elements, the columns of A (the vectors in the set) are linearly independent. Therefore, the set (a) is a spanning set for \(\mathbb{R}^{3}\).
02

Set (b)

The set (b) consists of four vectors: \((1,0,0)^{T},(0,1,1)^{T},(1,0,1)^{T},(1,2,3)^{T}\). Since there are more than three vectors in the set, they cannot be linearly independent, and therefore, the set (b) is not a spanning set for \(\mathbb{R}^{3}\).
03

Set (c)

The set (c) contains the vectors \((2,1,-2)^{T},(3,2,-2)^{T},(2,2,0)^{T}\). Let's create a matrix C with these vectors as columns: \[C = \begin{pmatrix} 2 & 3 & 2 \\ 1 & 2 & 2 \\ -2 & -2 & 0 \\ \end{pmatrix} \] We will now find the RREF of C: \[RREF(C) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ \end{pmatrix} \] Since the RREF of C has only 2 pivot elements, the columns of C are not linearly independent. Therefore, the set (c) is not a spanning set for \(\mathbb{R}^{3}\).
04

Set (d)

The set (d) consists of the vectors \((2,1,-2)^{T},(-2,-1,2)^{T},(4,2,-4)^{T}\). Let's create a matrix D with these vectors as columns: \[D = \begin{pmatrix} 2 & -2 & 4 \\ 1 & -1 & 2 \\ -2 & 2 & -4 \\ \end{pmatrix} \] Let's find the RREF of D: \[RREF(D) = \begin{pmatrix} 1 & -1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} \] Since the RREF of D has only 1 pivot element, the columns of D are not linearly independent. Therefore, the set (d) is not a spanning set for \(\mathbb{R}^{3}\).
05

Set (e)

Lastly, the set (e) contains two vectors: \((1,1,3)^{T},(0,2,1)^{T}\). Since there are only two vectors in the set, they cannot span \(\mathbb{R}^{3}\), so set (e) is not a spanning set for \(\mathbb{R}^{3}\). In conclusion, only set (a) is a spanning set for \(\mathbb{R}^{3}\).

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Most popular questions from this chapter

In each of the following, find the dimension of the subspace of \(P_{3}\) spanned by the given vectors: (a) \(x, x-1, x^{2}+1\) (b) \(x, x-1, x^{2}+1, x^{2}-1\) (c) \(x^{2}, x^{2}-x-1, x+1\) (d) \(2 x, x-2\)

In each of the following, determine the dimension of the subspace of \(\mathbb{R}^{3}\) spanned by the given vectors: (a) \(\left(\begin{array}{r}1 \\ -2 \\\ 2\end{array}\right),\left(\begin{array}{r}2 \\ -2 \\\ 4\end{array}\right),\left(\begin{array}{r}-3 \\ 3 \\ 6\end{array}\right)\) (b) \(\left(\begin{array}{l}1 \\ 1 \\\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 2 \\\ 3\end{array}\right),\left(\begin{array}{l}2 \\ 3 \\ 1\end{array}\right)\) (c) \(\left(\begin{array}{r}1 \\ -1 \\\ 2\end{array}\right),\left(\begin{array}{r}-2 \\ 2 \\\ -4\end{array}\right),\left(\begin{array}{r}3 \\ -2 \\\ 5\end{array}\right),\left(\begin{array}{r}2 \\ -1 \\ 3\end{array}\right)\)

Determine whether the following vectors are linearly independent in \(\mathbb{R}^{2}:\) (a) \(\left(\begin{array}{l}2 \\ 1\end{array}\right),\left(\begin{array}{l}3 \\\ 2\end{array}\right)\) (b) \(\left(\begin{array}{l}2 \\ 3\end{array}\right),\left(\begin{array}{l}4 \\\ 6\end{array}\right)\) (c) \(\left(\begin{array}{r}-2 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\\ 3\end{array}\right),\left(\begin{array}{l}2 \\ 4\end{array}\right)\) (d) \(\left(\begin{array}{r}-1 \\ 2\end{array}\right),\left(\begin{array}{r}1 \\\ -2\end{array}\right),\left(\begin{array}{r}2 \\ -4\end{array}\right)\) (e) \(\left(\begin{array}{l}1 \\ 2\end{array}\right),\left(\begin{array}{r}-1 \\\ 1\end{array}\right)\)

Let \(\mathbf{a}_{1}\) and \(\mathbf{a}_{2}\) be linearly independent vectors in \(\mathbb{R}^{3}\) and let \(\mathbf{x}\) be a vector in \(\mathbb{R}^{2}\) (a) Describe geometrically \(\operatorname{Span}\left(\mathbf{a}_{1}, \mathbf{a}_{2}\right)\) (b) If \(A=\left(\mathbf{a}_{1}, \mathbf{a}_{2}\right)\) and \(\mathbf{b}=A \mathbf{x},\) then what is the dimension of \(\operatorname{Span}\left(\mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{b}\right) ?\) Explain.

Determine the null space of each of the following matrices: $$\text { (a) }\left[\begin{array}{ll} 2 & 1 \\ 3 & 2 \end{array}\right]$$ $$\text { (b) }\left[\begin{array}{rrrr} 1 & 2 & -3 & -1 \\ -2 & -4 & 6 & 3 \end{array}\right]$$ $$\text { (c) }\left[\begin{array}{rrr} 1 & 3 & -4 \\ 2 & -1 & -1 \\ -1 & -3 & 4 \end{array}\right]$$ $$\text { (d) }\left[\begin{array}{rrrr} 1 & 1 & -1 & 2 \\ 2 & 2 & -3 & 1 \\ -1 & -1 & 0 & -5 \end{array}\right]$$
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