Activity 2.4.2

We would like to develop a means to detect when a set of vectors is linearly dependent. This activity will point the way.

1. Suppose we have five vectors in $\real^4$ that form the columns of a matrix having reduced row echelon form $$\left[\begin{array}{rrrrr} \vvec_1 & \vvec_2 & \vvec_3 & \vvec_4 & \vvec_5 \end{array}\right] \sim \left[\begin{array}{rrrrr} 1 & 0 & -1 & 0 & 2 \\ 0 & 1 & 2 & 0 & 3 \\ 0 & 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array}\right]$$. Is it possible to write one of the vectors $\vvec_1,\vvec_2,\ldots,\vvec_5$ as a linear combination of the others? If so, show explicitly how one vector appears as a linear combination of some of the other vectors. Is this set of vectors linearly dependent or independent?

2. Suppose we have another set of three vectors in $\real^4$ that form the columns of a matrix having reduced row echelon form $$\left[\begin{array}{rrr} \wvec_1 & \wvec_2 & \wvec_3 \\ \end{array}\right] \sim \left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{array}\right]$$. Is it possible to write one of these vectors $\wvec_1$, $\wvec_2$, $\wvec_3$ as a linear combination of the others? If so, show explicitly how one vector appears as a linear combination of some of the other vectors. Is this set of vectors linearly dependent or independent?

3. By looking at the pivot positions, how can you determine whether the columns of a matrix are linearly dependent or independent?

4. If one vector in a set is the zero vector $\zerovec$, can the set of vectors be linearly independent?

5. Suppose a set of vectors in $\real^{10}$ has twelve vectors. Is it possible for this set to be linearly independent?