# TBIL Activities for Understanding Linear Algebra

## Section2.4Linear independence

We start these activities by studying the span of two different sets of vectors.

### Activity2.4.0.1.

Consider the following vectors in $$\real^3\text{:}$$
\begin{equation*} \vvec_1=\threevec{0}{-1}{2}, \vvec_2=\threevec{3}{1}{-1}, \vvec_3=\threevec{2}{0}{1}\text{.} \end{equation*}
Describe the span of these vectors, $$\laspan{\vvec_1,\vvec_2,\vvec_3}\text{.}$$



### Activity2.4.0.2.

We will now consider a set of vectors that looks very much like the first set:
\begin{equation*} \wvec_1=\threevec{0}{-1}{2}, \wvec_2=\threevec{3}{1}{-1}, \wvec_3=\threevec{3}{0}{1}\text{.} \end{equation*}

#### (a)

Describe the span of these vectors, $$\laspan{\wvec_1,\wvec_2,\wvec_3}\text{.}$$



#### (b)

Show that the vector $$\wvec_3$$ is a linear combination of $$\wvec_1$$ and $$\wvec_2$$ by finding weights such that
\begin{equation*} \wvec_3 = a\wvec_1 + b\wvec_2\text{.} \end{equation*}

#### (c)

Using your work from the previous part, show how any any linear combination of $$\wvec_1\text{,}$$ $$\wvec_2\text{,}$$ and $$\wvec_3\text{,}$$
\begin{equation*} c_1\wvec_1 + c_2\wvec_2 + c_3\wvec_3 \end{equation*}
can be written as a linear combination of $$\wvec_1$$ and $$\wvec_2\text{.}$$

#### (d)

Explain why
\begin{equation*} \laspan{\wvec_1,\wvec_2,\wvec_3} = \laspan{\wvec_1,\wvec_2}\text{.} \end{equation*}

### Definition2.4.0.1.

A set of vectors $$\vvec_1,\vvec_2,\ldots,\vvec_n$$ is called linearly dependent if one of the vectors is a linear combination of the others. Otherwise, the set of vectors is called linearly independent.
We would like to develop a means of detecting when a set of vectors is linearly dependent. These questions will point the way.

### Activity2.4.0.3.

Consider whether the set of Euclidean vectors $$\left\{ \left[\begin{array}{c}-4\\2\\3\\0\\-1\end{array}\right], \left[\begin{array}{c}1\\2\\0\\0\\3\end{array}\right], \left[\begin{array}{c}1\\10\\10\\2\\6\end{array}\right], \left[\begin{array}{c}3\\4\\7\\2\\1\end{array}\right] \right\}$$ 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.

### Activity2.4.0.4.

Suppose we have five vectors in $$\real^4$$ that form the columns of a matrix having reduced row echelon form
\begin{equation*} \left[\begin{array}{rrrrr} \vvec_1 \amp \vvec_2 \amp \vvec_3 \amp \vvec_4 \amp \vvec_5 \end{array}\right] \sim \left[\begin{array}{rrrrr} 1 \amp 0 \amp -1 \amp 0 \amp 2 \\ 0 \amp 1 \amp 2 \amp 0 \amp 3 \\ 0 \amp 0 \amp 0 \amp 1 \amp -1 \\ 0 \amp 0 \amp 0 \amp 0 \amp 0 \\ \end{array}\right]\text{.} \end{equation*}

#### (a)

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.

#### (b)

Is this set of vectors linearly dependent or independent?

### Activity2.4.0.5.

Suppose we have another set of three vectors in $$\real^4$$ that form the columns of a matrix having reduced row echelon form
\begin{equation*} \left[\begin{array}{rrr} \wvec_1 \amp \wvec_2 \amp \wvec_3 \\ \end{array}\right] \sim \left[\begin{array}{rrr} 1 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \\ 0 \amp 0 \amp 0 \\ \end{array}\right]\text{.} \end{equation*}

#### (a)

Is it possible to write one of these vectors $$\wvec_1\text{,}$$ $$\wvec_2\text{,}$$ $$\wvec_3$$ 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.

#### (b)

Is this set of vectors linearly dependent or independent?

#### (c)

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

### Activity2.4.0.6.

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

### Activity2.4.0.7.

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

### Activity2.4.0.8.

Let $$\vvec_1,\vvec_2,\vvec_3$$ be vectors in $$\mathbb R^n\text{.}$$ Suppose $$3\vvec_1-5\vvec_2=\vvec_3\text{,}$$ so the set $$\{\vvec_1,\vvec_2,\vvec_3\}$$ is linearly dependent. Which of the following is true of the vector equation $$x_1\vvec_1+x_2\vvec_2+x_3\vvec_3=\zerovec\text{?}$$
1. It is consistent with one solution.
2. It is consistent with infinitely many solutions.
3. It is inconsistent.

### Activity2.4.0.9.

We now want to explore the connection between linear independence and homogeneous equations.

#### (a)

Explain why the homogenous matrix equation $$A\xvec = \zerovec$$ is consistent no matter the matrix $$A\text{.}$$

#### (b)

Consider the matrix
\begin{equation*} A = \left[\begin{array}{rrr} 3 \amp 2 \amp 0 \\ -1 \amp 0 \amp -2 \\ 2 \amp 1 \amp 1 \end{array}\right] \end{equation*}
whose columns we denote by $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3\text{.}$$ Are the vectors $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3$$ linearly dependent or independent?



#### (c)

Give a parametric description (in set notation) of the solution space of the homogeneous equation $$A\xvec = \zerovec\text{.}$$

#### (d)

We know that $$\zerovec$$ is a solution to the homogeneous equation. Find another solution that is different from $$\zerovec\text{.}$$ Use your solution to find weights $$c_1\text{,}$$ $$c_2\text{,}$$ and $$c_3$$ such that
\begin{equation*} c_1\vvec_1 + c_2\vvec_2 + c_3\vvec_3 = \zerovec\text{.} \end{equation*}

#### (e)

Use the expression you found in the previous part to write one of the vectors as a linear combination of the others.

### Activity2.4.0.10.

What is the largest number of $$\IR^4$$ vectors that can form a linearly independent set?
1. $$\displaystyle 3$$
2. $$\displaystyle 4$$
3. $$\displaystyle 5$$
4. You can have infinitely many vectors and still be linearly independent.