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Reflections 20.7 Reflect on your understanding
In addition to the reflection activities below, re-read
Section 20.2 Terminology and notation . Be sure you understand each of the new definitions introduced in this chapter, and spend some time committing them to memory.
1. Domain of a column space.
The column space of an
\(m \times n \) matrix is always a subspace of
\(\R^k \) for some specific value of
\(k \text{.}\) What determines the value of
\(k \text{?}\)
2. Column space versus row equivalence.
3. Row equivalence versus column dependence.
(a) If
\(A \) and
\(B \) are
row equivalent matrices , are the relationships of linear dependence and independence between columns of
\(A \) always the same as or sometimes different from the relationships of linear dependence and independence between the corresponding columns of
\(B \text{?}\)
(b) Explain how to use row reduction to test a set of vectors for independence.
(c) Explain how to use row reduction to determine an expression for one vector as a linear combination of a set of vectors.
4. Column space basis procedure.
For a matrix
\(A \text{:}\)
(a) State the connection between the pattern of positions of leading ones in
\(\RREF(A) \) and the choice of basis vectors for the column space of
\(A \text{.}\)
(b) Explain why we choose our column space basis vectors from amongst the columns of
\(A \) and not
\(\RREF(A) \text{.}\)
5. Domain of a row space.
The row space of an
\(m \times n \) matrix is always a subspace of
\(\R^k \) for some specific value of
\(k \text{.}\) What determines the value of
\(k \text{?}\)
6. Row space versus row equivalence.
7. Row equivalence versus row dependence.
If
\(A \) and
\(B \) are
row equivalent matrices , are the relationships of linear dependence and independence between rows of
\(A \) always the same as or sometimes different from the relationships of linear dependence and independence between the corresponding rows of
\(B \text{?}\)
8. Row independence in RREF.
Explain why the pattern of leading ones in an RREF matrix always guarantees that the nonzero rows form a linearly independent set.
9. Row space basis procedure.
For a matrix
\(A \text{:}\)
(a) State the connection between the pattern of positions of leading ones in
\(\RREF(A) \) and the choice of basis vectors for the row space of
\(A \text{.}\)
(b) Explain why we choose our row space basis vectors from amongst the rows of
\(\RREF(A) \) and not
\(A \text{.}\)
10. Comparison of spanning sets via row reduction.
Explain how to use row reduction to determine whether two sets of vectors span the same space.
11. Column space procedure versus row space procedure.
Given a spanning set for a vector space, explain the relative advantages of each of the column space procedure and the row space procedure in obtaining a basis for the space.
12. Domain of a null space.
The null space of an
\(m \times n \) matrix is always a subspace of
\(\R^k \) for some specific value of
\(k \text{.}\) What determines the value of
\(k \text{?}\)
13. Null space versus row equivalence.
14. Null space procedure.
Explain the procedure to determine a basis for the null space of a matrix from parametric equations expressing the general solution to the associated homogeneous system.
15. Dimensions of the three spaces.
For each of the column, row, and null spaces, explain the connection between the rank of a matrix and the dimension of that space.
