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Section 3.5 Subspaces of \(\real^p\)
Activity 3.5.1 .
Let’s consider the following matrix \(A\) and its reduced row echelon form.
\begin{equation*}
A = \left[\begin{array}{rrrr}
2 \amp -1 \amp 2 \amp 3 \\
1 \amp 0 \amp 0 \amp 2 \\
-2 \amp 2 \amp -4 \amp -2 \\
\end{array}\right]
\sim
\left[\begin{array}{rrrr}
1 \amp 0 \amp 0 \amp 2 \\
0 \amp 1 \amp -2 \amp 1 \\
0 \amp 0 \amp 0 \amp 0 \\
\end{array}\right]\text{.}
\end{equation*}
(a) Are the columns of
\(A\) linearly independent? Do they span
\(\real^3\text{?}\)
(b) Give a parametric description of the solution space to the homogeneous equation
\(A\xvec = \zerovec\text{.}\)
(c) Explain how this parametric description produces two vectors
\(\wvec_1\) and
\(\wvec_2\) whose span is the solution space to the equation
\(A\xvec = \zerovec\text{.}\)
(d) What can you say about the linear independence of the set of vectors
\(\wvec_1\) and
\(\wvec_2\text{?}\)
(e) Let’s denote the columns of
\(A\) as
\(\vvec_1\text{,}\) \(\vvec_2\text{,}\) \(\vvec_3\text{,}\) and
\(\vvec_4\text{.}\) Explain why
\(\vvec_3\) and
\(\vvec_4\) can be written as linear combinations of
\(\vvec_1\) and
\(\vvec_2\text{.}\)
(f) Explain why
\(\vvec_1\) and
\(\vvec_2\) are linearly independent and
\(\laspan{\vvec_1,\vvec_2} =
\laspan{\vvec_1, \vvec_2, \vvec_3, \vvec_4}\text{.}\)
Definition 3.5.1 .
A
subspace of
\(\real^p\) is a nonempty subset of
\(\real^p\) such that any linear combination of vectors in that set is also in the set.
Activity 3.5.2 .
We will look at some subspaces of
\(\real^2\text{.}\)
(a) Explain why a line that does not pass through the origin is not a subspace of
\(\real^2\text{.}\)
(b) Explain why any subspace of
\(\real^2\) must contain the zero vector
\(\zerovec\text{.}\)
(c) Explain why the subset
\(S\) of
\(\real^2\) that consists of only the zero vector
\(\zerovec\) is a subspace of
\(\real^2\text{.}\)
(d) Explain why the subspace
\(S=\real^2\) is itself a subspace of
\(\real^2\text{.}\)
(e) If
\(\vvec\) and
\(\wvec\) are two vectors in a subspace
\(S\text{,}\) explain why
\(\laspan{\vvec,\wvec}\) is contained in the subspace
\(S\) as well.
Definition 3.5.2 .
A
basis for a subspace
\(S\) of
\(\real^p\) is a set of vectors in
\(S\) that are linearly independent and span
\(S\text{.}\) It can be seen that any two bases have the same number of vectors. Therefore, we say that the
dimension of the subspace
\(S\text{,}\) denoted
\(\dim
S\text{,}\) is the number of vectors in any basis.
Definition 3.5.3 .
If
\(A\) is an
\(m\times n\) matrix, we call the subset of vectors
\(\xvec\) in
\(\real^n\) satisfying
\(A\xvec = \zerovec\) the
null space of
\(A\text{.}\) We denote it as
\(\nul(A)\text{.}\)
We will explore some null spaces in the next couple of activities.
Activity 3.5.3 .
Consider the matrix
\begin{equation*}
A=\left[\begin{array}{rrr}
1 \amp 3 \amp -1 \\
-2 \amp 0 \amp -4 \\
1 \amp 2 \amp 0 \\
\end{array}\right]
\end{equation*}
(a) Give a parametric description of the null space
\(\nul(A)\text{.}\)
(b) Give a basis for and state the dimension of
\(\nul(A)\text{.}\)
(c) The null space
\(\nul(A)\) is a subspace of
\(\real^p\) for which
\(p\text{?}\)
Activity 3.5.4 .
Consider the matrix \(A\) whose reduced row echelon form is given:
\begin{equation*}
A \sim
\left[\begin{array}{rrrr}
1 \amp 2 \amp 0 \amp -3 \\
0 \amp 0 \amp 1 \amp 2 \\
\end{array}\right]\text{.}
\end{equation*}
(a) Give a parametric description of
\(\nul(A)\text{.}\)
(b) Notice that the parametric description gives a set of vectors that span
\(\nul(A)\text{.}\) Explain why this set of vectors is linearly independent and hence forms a basis. What is the dimension of
\(\nul(A)\text{?}\)
(c) For this matrix,
\(\nul(A)\) is a subspace of
\(\real^p\) for what
\(p\text{?}\)
Activity 3.5.5 .
(a) What is the relationship between the dimensions of the matrix
\(A\text{,}\) the number of pivot positions of
\(A\) and the dimension of
\(\nul(A)\text{?}\)
(b) Suppose that the columns of a matrix
\(A\) are linearly independent. What can you say about
\(\nul(A)\text{?}\)
(c) If
\(A\) is an invertible
\(n\times n\) matrix, what can you say about
\(\nul(A)\text{?}\)
Activity 3.5.6 .
Suppose that
\(A\) is a
\(5\times 10\) matrix and that
\(\nul(A) = \real^{10}\text{.}\) What can you say about the matrix
\(A\text{?}\)
Definition 3.5.4 .
The
rank of a matrix
\(A\text{,}\) denoted
\(\rank(A)\text{,}\) is the number of pivot positions of
\(A\text{.}\)
Definition 3.5.5 .
If
\(A\) is an
\(m\times n\) matrix, we call the span of its columns the
column space of
\(A\) and denote it as
\(\col(A)\text{.}\)
We will explore some column spaces in the next couple of activities.
Activity 3.5.7 .
Consider the matrix
\begin{equation*}
A= \left[\begin{array}{rrr}
\vvec_1 \amp \vvec_2 \amp \vvec_3
\end{array}\right]
=
\left[\begin{array}{rrr}
1 \amp 3 \amp -1 \\
-2 \amp 0 \amp -4 \\
1 \amp 2 \amp 0 \\
\end{array}\right]\text{.}
\end{equation*}
(a) Since
\(\col(A)\) is the span of the columns, the vectors
\(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and
\(\vvec_3\) naturally span
\(\col(A)\text{.}\) Are these vectors linearly independent?
(b) Show that
\(\vvec_3\) can be written as a linear combination of
\(\vvec_1\) and
\(\vvec_2\) by giving the linear combination. Then explain why
\(\col(A)=\laspan{\vvec_1,\vvec_2}\text{.}\)
(c) Explain why the vectors
\(\vvec_1\) and
\(\vvec_2\) form a basis for
\(\col(A)\text{.}\) This shows that
\(\col(A)\) is a 2-dimensional subspace of
\(\real^3\) and is therefore a plane in
\(\real^3\text{.}\)
Activity 3.5.8 .
Consider the matrix \(A\) and its reduced row echelon form:
\begin{equation*}
A = \left[\begin{array}{rrrr}
-2 \amp -4 \amp 0 \amp 6 \\
1 \amp 2 \amp 0 \amp -3 \\
\end{array}\right]
\sim
\left[\begin{array}{rrrr}
1 \amp 2 \amp 0 \amp -3 \\
0 \amp 0 \amp 0 \amp 0 \\
\end{array}\right]\text{.}
\end{equation*}
We will call the columns \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) \(\vvec_3\text{,}\) and \(\vvec_4\text{.}\)
(a) Explain why
\(\vvec_2\text{,}\) \(\vvec_3\text{,}\) and
\(\vvec_4\) can be written as a linear combination of
\(\vvec_1\text{.}\)
(b) Explain why
\(\col(A)\) is a 1-dimensional subspace of
\(\real^2\) and is therefore a line.
(c) What is the relationship between the dimension
\(\dim~\col(A)\) and the rank
\(\rank(A)\text{?}\)
(d) What is the relationship between the dimension of the column space
\(\col(A)\) and the null space
\(\nul(A)\text{?}\)
Activity 3.5.9 .
If
\(A\) is an invertible
\(9\times9\) matrix, what can you say about the column space
\(\col(A)\text{?}\)
Activity 3.5.10 .
If
\(\col(A)=\{\zerovec\}\text{,}\) what can you say about the matrix
\(A\text{?}\)
Activity 3.5.11 .
Consider a matrix
\(A\text{.}\) Which of the following is equal to the dimension of the null space of
\(A\text{?}\)
The number of pivot columns
The number of non-pivot columns
The number of non-pivot rows
Activity 3.5.12 .
Consider a matrix
\(A\text{.}\) Which of the following is equal to the dimension of the column space of
\(A\text{?}\)
The number of pivot columns
The number of non-pivot columns
The number of non-pivot rows
Activity 3.5.13 .
Consider the matrix
\begin{equation*}
A
=
\left[\begin{array}{ccc} 1 & -3 & 2\\ 2 & -6 & 0 \\ 0 & 0 & 1 \\ -1 & 3 & 1 \end{array}\right]
.
\end{equation*}
Verify that if \(A\) is an \(m\times n\) matrix, then
\begin{equation*}
\dim~\nul(A) + \dim~\col(A) = n
\end{equation*}
holds for \(A\text{.}\) This equation is also known as the Rank-Nullity Theorem.
