Linear Independence (V5)

Section 2.5 Linear Independence (V5)

Activity 2.5.1.

Consider the two sets

S = {[\begin{matrix} 2 \\ 3 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 1 \\ 4 \end{matrix}]} T = {[\begin{matrix} 2 \\ 3 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 1 \\ 4 \end{matrix}], [\begin{matrix} - 1 \\ 0 \\ - 11 \end{matrix}]} .

Which of the following is true?

$span S$ is bigger than $span T .$
$span S$ and $span T$ are the same size.
$span S$ is smaller than $span T .$

🔗

Definition 2.5.2.

We say that a set of vectors is linearly dependent if one vector in the set belongs to the span of the others. Otherwise, we say the set is linearly independent.

Figure 2.5.3. A linearly dependent set of three vectors

You can think of linearly dependent sets as containing a redundant vector, in the sense that you can drop a vector out without reducing the span of the set. In the above image, all three vectors lay in the same planar subspace, but only two vectors are needed to span the plane, so the set is linearly dependent.

🔗

Activity 2.5.4.

Let ${\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}$ be vectors in $R^{n} .$ Suppose $3 {\vec{v}}_{1} - 5 {\vec{v}}_{2} = {\vec{v}}_{3},$ so the set ${{\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}}$ is linearly dependent. Which of the following is true of the vector equation $x_{1} {\vec{v}}_{1} + x_{2} {\vec{v}}_{2} + x_{3} {\vec{v}}_{3} = \vec{0}$ ?

It is consistent with one solution
It is consistent with infinitely many solutions
It is inconsistent.

🔗

Fact 2.5.5.

For any vector space, the set ${{\vec{v}}_{1}, \dots {\vec{v}}_{n}}$ is linearly dependent if and only if the vector equation $x_{1} {\vec{v}}_{1} + \dots + x_{n} {\vec{v}}_{n} = \vec{0}$ is consistent with infinitely many solutions.

🔗

Activity 2.5.6.

Find

RREF [\begin{array}{cccccc} 2 & 2 & 3 & - 1 & 4 & 0 \\ 3 & 0 & 13 & 10 & 3 & 0 \\ 0 & 0 & 7 & 7 & 0 & 0 \\ - 1 & 3 & 16 & 14 & 1 & 0 \end{array}]

and mark the part of the matrix that demonstrates that

S = {[\begin{matrix} 2 \\ 3 \\ 0 \\ - 1 \end{matrix}], [\begin{matrix} 2 \\ 0 \\ 0 \\ 3 \end{matrix}], [\begin{matrix} 3 \\ 13 \\ 7 \\ 16 \end{matrix}], [\begin{matrix} - 1 \\ 10 \\ 7 \\ 14 \end{matrix}], [\begin{matrix} 4 \\ 3 \\ 0 \\ 1 \end{matrix}]}

is linearly dependent (the part that shows its linear system has infinitely many solutions).

🔗

Observation 2.5.7.

A set of Euclidean vectors ${{\vec{v}}_{1}, \dots {\vec{v}}_{n}}$ is linearly dependent if and only if $RREF [\begin{array}{ccc} {\vec{v}}_{1} & \dots & {\vec{v}}_{n} \end{array}]$ has a column without a pivot position.

🔗

Observation 2.5.8.

Compare the following results:

A set of $R^{m}$ vectors ${{\vec{v}}_{1}, \dots {\vec{v}}_{n}}$ is linearly independent if and only if $RREF [\begin{array}{ccc} {\vec{v}}_{1} & \dots & {\vec{v}}_{n} \end{array}]$ has all pivot columns.
A set of $R^{m}$ vectors ${{\vec{v}}_{1}, \dots {\vec{v}}_{n}}$ spans $R^{m}$ if and only if $RREF [\begin{array}{ccc} {\vec{v}}_{1} & \dots & {\vec{v}}_{n} \end{array}]$ has all pivot rows.

🔗

Activity 2.5.9.

Consider whether the set of Euclidean vectors ${[\begin{matrix} - 4 \\ 2 \\ 3 \\ 0 \\ - 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 0 \\ 0 \\ 3 \end{matrix}], [\begin{matrix} 1 \\ 10 \\ 10 \\ 2 \\ 6 \end{matrix}], [\begin{matrix} 3 \\ 4 \\ 7 \\ 2 \\ 1 \end{matrix}]}$ is linearly dependent or linearly independent.

🔗

(a)

Reinterpret this question as an appropriate question about solutions to a vector equation.

🔗

(b)

Use the solution to this question to answer the original question.

🔗

Activity 2.5.10.

Consider whether the set of polynomials ${x^{3} + 1, x^{2} + 2 x, x^{2} + 7 x + 4}$ is linearly dependent or linearly independent.

🔗

(a)

Reinterpret this question as an appropriate question about solutions to a polynomial equation.

🔗

(b)

Use the solution to this question to answer the original question.