- (a)
- Is in YES, NO
Suppose is in . Then we can write \begin{align*} \vec{w} &= a_1 (\vec{v}_{1}+\vec{w}) + \dots + a_m (\vec{v}_{m}+\vec{w}) \\ \vec{w} &= a_1\vec{v}_{1} + \dots + a_m\vec{v}_{m}+ (a_1 + \dots + a_m)\vec{w}. \end{align*}
Now consider two cases separately: either or . In either case, arrive at a contradiction and conclude that is not in .
- (b)
- Is linearly independent? YES, NO
Challenge Problems for Chapter 3
Suppose is a linearly independent set in , and that is not in .
Suppose , , and are the rows of the matrix . Then we can interpret the solution to
the system of equations as the intersection of three planes containing the origin.
Discuss what this intersection would look like geometrically if the reduced row
echelon form of is of the form:
- (a)
- \begin{equation*} \left [ \begin{array}{ccc|c} 1 & 0 & * & 0 \\ 0 & 1 & * & 0 \\ 0 & 0 & 0 & 0 \end{array} \right ] \end{equation*}
- (b)
- \begin{equation*} \left [ \begin{array}{ccc|c} 1 & * & * & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right ] \end{equation*}
- (c)
- Are there any other possibilities?
Show that for any matrix , has a row of zeros if and only if one of the rows of can
be expressed as a linear combination of the others.
Recall that a plane in has an equation of the form , where , , are the components of
a normal vector to the plane. Suppose three planes intersect in a line, as shown
below.
We can find the line of intersection by solving a system of equations. Suppose the augmented matrix corresponds to this system.
- (a)
- Explain why will have a row of zeros.
- (b)
- Consider the normal vectors to the three planes. What geometric property of these particular three normal vectors explains the row of zeros in the reduced row-echelon form?
Bibliography
The first problem came from the end of Chapter 1 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)
W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, pp. 33–34.
2024-09-11 17:54:56