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 \(\laspan{\vvec_1,\vvec_2} =
\laspan{\vvec_1, \vvec_2, \vvec_3, \vvec_4}\text{.}\)
Definition3.5.0.1.
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.
Activity3.5.0.2.
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.
Definition3.5.0.2.
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.
Definition3.5.0.3.
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.
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 \(p\text{?}\)
Activity3.5.0.5.
(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 \(\nul(A)\text{?}\)
Activity3.5.0.6.
Suppose that \(A\) is a \(5\times 10\) matrix and that \(\nul(A) = \real^{10}\text{.}\) What can you say about the matrix \(A\text{?}\)
Definition3.5.0.4.
The rank of a matrix \(A\text{,}\) denoted \(\rank(A)\text{,}\) is the number of pivot positions of \(A\text{.}\)
Definition3.5.0.5.
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.
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 \(\real^3\text{.}\)
Activity3.5.0.8.
Consider the matrix \(A\) and its reduced row echelon form:
