Linear Independence

Determine which of the following sets are linearly independent. For those sets that are linearly dependent, write one of the vectors as a linear combination of the others.

Consider the matrix $\mathbf{A} = \begin{pmatrix} 2 & 1 & 1 & 0 \\ 4 & 2 & 1 & 2 \\ 6 & 3 & 2 & 2 \end{pmatrix}$.

Suppose that in a population of a million children the height of each one is measured at ages 1 year, 2 years, and 3 years, and accumulate this data in a matrix

= H.

Explain why there are at most three independent children'' in the sense that the heights of all the other children must be a combination of these independent'' ones.

Consider a particular species of wildflower in which each plant has several stems, leaves, and flowers, and for each plant let the following hold. $S =$ the average stem length (in inches). $L =$ the average leaf width (in inches). $F =$ the number of flowers. Four particular plants are examined, and the information is tabulated in the following matrix:

= A.

For these four plants, determine whether or not there exists a linear relationship between $S$, $L$, and $F$. In other words, do there exist constants $\alpha_0, \alpha_1, \alpha_2$, and $\alpha_3$ such that $\alpha_0 + \alpha_1 S + \alpha_2 L + \alpha_3 F = 0$?

Let $\mathcal{S} = \left\{ \overrightarrow {0}\right\}$ be the set containing only the zero vector.

If $\mathbf{T}$ is a triangular matrix in which each $t_{ii} \neq 0$, explain why the rows and columns of $\mathbf{T}$ must each be linearly independent sets.

Determine whether or not the following set of matrices is a linearly independent set:

Without doing any computation, determine whether the following matrix is singular or nonsingular:

In theory, determining whether or not a given set is linearly independent is a well-defined problem with a straightforward solution. In practice, however, this problem is often not so well defined because it becomes clouded by the fact that we usually cannot use exact arithmetic, and contradictory conclusions may be produced depending upon the precision of the arithmetic. For example, let

If $\mathbf{A}_{m \times n}$ is a matrix such that $\sum_{j=1}^{n} a_{ij} = 0$ for each $i = 1, 2, \ldots , m$ (i.e., each row sum is $0$), explain why the columns of $\mathbf{A}$ are a linearly dependent set, and hence $\operatorname {rk}\left(\mathbf{A}\right) < n$.

If $\mathcal{S} = \left\{ \overrightarrow {u}_1, \overrightarrow {u}_2, \ldots , \overrightarrow {u}_n\right\}$ is a linearly independent subset of $\mathbb {R}^{m \times 1}$, and if $\mathbf{P}_{m \times m}$ is a nonsingular matrix, explain why the set

must also be a linearly independent set. Is this result still true if $\mathbf{P}$ is singular?

Suppose that $\mathcal{S} = \left\{ \overrightarrow {u}_1, \overrightarrow {u}_2, \ldots , \overrightarrow {u}_n\right\}$ is a set of vectors from $\mathbb {R}^{m}$. Prove that $\mathcal{S}$ is linearly independent if and only if the set

is linearly independent.

Which of the following sets of functions are linearly independent?

Prove that the converse of the statement given in Example 4.3.6 is false by showing that $\mathcal{S} = \left\{ x^3, \; \left|x\right|^3\right\}$ is a linearly independent set, but the associated Wronski matrix $\mathbf{W}(x)$ is singular for all values of $x$.

Example 4.3.6, referenced above: if a set of functions is linearly dependent on an interval, then its Wronski matrix is singular at every point of that interval.

If $\mathbf{A}^T$ is diagonally dominant, explain why partial pivoting is not needed when solving $\mathbf{A}\overrightarrow {x} = \overrightarrow {b}$ by Gaussian elimination.