Objectives: Topics
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Subspaces, column space, and null space.
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A basis for a subspace.
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Section 2.8 Subspaces
Handout 2.8 Subspaces
Objectives: Goals
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Determine whether a set is a subspace.
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Determine whether a vector is in a particular subspace, or find a vector in that subspace.
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Construct a basis for a subspace (e.g., a basis for \(\operatorname{Col}(A)\)).
Motivating Question: Given a matrix \(A\text{,}\) what is the set of vectors \(\vec b\) for which we can solve \(A\vec x=\vec b\text{?}\)
Definition 2.43. Subset of \(\mathbb{R}^n\).
Definition 2.44. Subspace.
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\(\displaystyle \vec 0 \in H\)
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\(\vec u+\vec v\in H\) for all \(\vec u,\vec v\in H\text{.}\)
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\(c\vec u\in H\) for any scalar \(c\in\mathbb{R}\) and any \(\vec u\in H\text{.}\)
Note that scalar closure implies the zero vector must be in \(H\text{.}\)
Example 2.45.
Which of the following subsets could be subspaces of \(\mathbb{R}^2\text{?}\)
Example 2.46.
For each subset of \(\mathbb{R}^3\) given below, determine if it is a subspace of \(\mathbb{R}^3\text{.}\)
(a)
\(S_1 = \left\{\vec 0\right\}\)
(b)
\(S_2 = \left\{\vec 0, \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix},\begin{bmatrix}-1 \\ 0 \\ -1\end{bmatrix} \right\}\)
(c)
\(S_3 = \left\{\begin{bmatrix}1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}\right\}\)
(d)
\(S_4 = \operatorname{Span}\left\{\begin{bmatrix}1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}\right\}\)
Example 2.47.
The following subsets of \(\mathbb{R}^2\) are written in set-builder notation. For each, determine if it is a subspace of \(\mathbb{R}^2\text{.}\) Note that \(\mathbb{Z} = \{\dots, -2, -1, 0, 1 , 2, \dots\}\) is the set of integers.
(a)
\(T_1 = \left\{\begin{bmatrix} x \\ -x \end{bmatrix} \in \mathbb{R}^2 \ \Big \rvert \ x\in\mathbb{Z}\right\}\)
(b)
\(T_2 = \left\{\begin{bmatrix} x \\ -x \end{bmatrix} \in \mathbb{R}^2 \ \Big \rvert \ x\in\mathbb{R}\right\}\)
(c)
\(T_3 = \left\{\vec x \in \mathbb{R}^2 \ \Big \rvert \ \begin{bmatrix} 1 \amp 2 \\ 2 \amp 4 \end{bmatrix}\vec x = \begin{bmatrix} 0 \\ 0 \end{bmatrix}\right\}\)
(d)
\(T_4 = \left\{\vec x \in \mathbb{R}^2 \ \Big \rvert \ \begin{bmatrix} 1 \amp 2 \\ 2 \amp 4 \end{bmatrix}\vec x = \begin{bmatrix} 1 \\ 2 \end{bmatrix}\right\}\)
Recall that for vectors \(\vec v_1,\dots,\vec v_p\in\mathbb{R}^n\text{,}\) \(\operatorname{Span}\{\vec v_1,\dots,\vec v_p\}\) is the subspace consisting of all linear combinations of these vectors. Not only is the span of vectors an example of a subspace of \(\mathbb{R}^n\text{,}\) they are the only examples of subspace of \(\mathbb{R}^n\text{.}\)
Theorem 2.48.
Every subspace of \(\mathbb{R}^n\) can be written as the span of a finite number of vectors; that is, if \(H\) is a subspace of \(\mathbb{R}^n\text{,}\) then there exists vectors \(\vec v_1, \dots, \vec v_k\) so that \(H = \operatorname{Span}\left\{\vec v_1, \dots, \vec v_n\right\}\text{.}\)
Definition 2.49. Column Space and Null Space.
Let \(A=\begin{bmatrix}\vec a_1 \amp \cdots \amp \vec a_n\end{bmatrix}\) be an \(m\times n\) matrix.
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The column space of \(A\text{,}\) \(\operatorname{Col}(A)\text{,}\) is the subspace of \(\mathbb{R}^m\) spanned by \(\vec a_1,\dots,\vec a_n\text{.}\)
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The null space of \(A\text{,}\) \(\operatorname{Nul}(A)\text{,}\) is the subspace of \(\mathbb{R}^n\) consisting of all solutions to \(A\vec x=\vec 0\text{.}\)
Theorem 2.50.
Let \(A\) be an \(m \times n\) matrix, and let \(B\) be any matrix which is row equivalent to \(A\text{.}\) Then
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The subspaces \(\operatorname{Col}(A)\) and \(\operatorname{Col}(B)\) are most likely different. So, often, \(\operatorname{Col}(A) \neq \operatorname{Col}(B)\text{.}\)
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The subspaces \(\operatorname{Nul}(A)\) and \(\operatorname{Nul}(B)\) are always the same. So, it is always true that \(\operatorname{Nul}(A) = \operatorname{Nul}(B)\text{.}\)
Example 2.51.
\begin{equation*}
A=
\begin{bmatrix}
1 \amp -3 \amp -4\\
-4 \amp 6 \amp -2\\
-3 \amp 7 \amp 6
\end{bmatrix}
\sim
\begin{bmatrix}
1 \amp 0 \amp 5\\
0 \amp 1 \amp 3\\
0 \amp 0 \amp 0
\end{bmatrix} = B,
\qquad
\vec b=
\begin{bmatrix}
3\\3\\-4
\end{bmatrix}.
\end{equation*}
Example 2.52.
Determine whether \(\vec v\) is in \(\operatorname{Nul}(A)\text{.}\)
\begin{equation*}
A = \begin{bmatrix}
1 \amp -3 \amp -4\\
-4 \amp 6 \amp -2\\
-3 \amp 7 \amp 6
\end{bmatrix} \sim
\begin{bmatrix}
1 \amp 0 \amp 5\\
0 \amp 1 \amp 2\\
0 \amp 0 \amp 0
\end{bmatrix} = B,
\qquad
\vec v=
\begin{bmatrix}
-5t\\
-3t\\
t
\end{bmatrix},
\quad t\in\mathbb{R}.
\end{equation*}
Definition 2.53. Basis.
A basis for a subspace \(H\) is a set of linearly independent vectors in \(H\) that span \(H\text{.}\)
In MATH 1554, each basis will be a finite set of vectors. For every subspace other than the trivial subspace \(\left\{\vec 0\right\}\text{,}\) there are infinitely many choices for the basis.
Be careful to NEVER include \(\Span\) when writing your basis. Writing \(\Span\) of the basis gives the entire subspace. A basis is NOT the entire subspace, but a finite set of building blocks used to build the space.
Example 2.54.
Let
\begin{equation*}
H=\left\{
\begin{bmatrix}
x_1\\x_2\\x_3\\x_4
\end{bmatrix}
\in\mathbb{R}^4\;\middle|\;
x_1+2x_2+x_3+5x_4=0
\right\}.
\end{equation*}
(a)
Find at least three vectors which live in \(H\text{.}\)
(b)
(c)
Construct a basis for \(H\text{.}\)
Example 2.55.
\begin{equation*}
A=
\begin{bmatrix}
-3 \amp 6 \amp -1 \amp 0\\
1 \amp -2 \amp 2 \amp 0\\
2 \amp -4 \amp 5 \amp 0
\end{bmatrix}
\sim
\begin{bmatrix}
1 \amp -2 \amp 0 \amp 0\\
0 \amp 0 \amp 1 \amp 0\\
0 \amp 0 \amp 0 \amp 0
\end{bmatrix} = B.
\end{equation*}
Example 2.56.
Let
\begin{equation*}
V=\left\{
\begin{bmatrix}
a\\b
\end{bmatrix}
\in\mathbb{R}^2\;\middle|\;ab=0
\right\}.
\end{equation*}
(a)
Give an example of a vector that is in \(V\text{.}\)
(b)
Give an example of a vector that is not in \(V\text{.}\)
(c)
Is the zero vector in \(V\text{?}\)
