Section 4.2 Independence and Dimension

Definition: An indexed set of vectors $\left\{\begin{array}{c}\overrightarrow{v_1},\cdots ,\overrightarrow{v_p}\end{array}\right\}$ in ${\mathbb{R}}^n$ is said to be linearly independent if the vector equation $x_1\overrightarrow{v_1},\cdots ,x_p\overrightarrow{v_p}=0$ in ${\mathbb{R}}^n$ has only trivial solution. $\left\{\begin{array}{c}\overrightarrow{v_1},\cdots ,\overrightarrow{v_p}\end{array}\right\}$ in ${\mathbb{R}}^n$ is said to be linearly dependent if there are $c_1,\cdots ,c_p$ not all zero such that $c_1\overrightarrow{v_1},\cdots ,c_p\overrightarrow{v_p}=0$. $c_1\overrightarrow{v_1},\cdots ,c_p\overrightarrow{v_p}=0$ is called the linear dependence relation among $\overrightarrow{v_1},\cdots ,\overrightarrow{v_p}$.

Example 1: Determine if the set $\left\{\begin{array}{c}\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}\end{array}\right\}$ is linearly independent. If possible, find a linear dependence relation among $\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}$. $\overrightarrow{v_1}=\left[\begin{array}{c}-1\\ 2\\ 3\end{array}\right]$, $\overrightarrow{v_2}=\left[\begin{array}{c}4\\ -1\\ 9\end{array}\right]$, and $\overrightarrow{v_3}=\left[\begin{array}{c}2\\ -4\\ -6\end{array}\right]$.

Exercise 1: Determine if the set $\left\{\begin{array}{c}\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}\end{array}\right\}$ is linearly independent. If possible, find a linear dependence relation among $\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}$. $\overrightarrow{v_1}=\left[\begin{array}{c}-2\\ 2\\ -3\end{array}\right]$, $\overrightarrow{v_2}=\left[\begin{array}{c}6\\ -1\\ 4\end{array}\right]$, and $\overrightarrow{v_3}=\left[\begin{array}{c}4\\ -4\\ 6\end{array}\right]$.

Note: 1. Given a matrix $A=\left[\begin{array}{c}\overrightarrow{v_1},\cdots ,\overrightarrow{v_p}\end{array}\right]$ with $p$ columns, the matrix equation $A\overrightarrow{x}=\overrightarrow{0}$ can be written as $x_1\overrightarrow{v_1}+\cdots +x_p\overrightarrow{v_p}=0$. Then each linear dependence relation among the columns of $A$ corresponds to a nontrivial solution of $A\overrightarrow{x}=\overrightarrow{0}$. Hence the columns of matrix $A$ are linearly independent if and only if the equation $A\overrightarrow{x}=0$ has only the trivial solution.

2. A set with only one vector is linearly independent if and only if it is not a zero vector. The zero vector is linearly dependent.

3. A set with two vectors is linearly independent if and only if they are not multiple of each others.

Example 2: Show the column set of $A$ is a linearly independent set. $A=\left[\begin{array}{ccc}2& 0& 1\\ 1& -1& 0\\ -1& 2& 1\end{array}\right]$.

Exercise 2: Show the column set of $A$ is a linearly independent set. $A=\left[\begin{array}{ccc}1& 2& 1\\ 0& -1& 2\\ -1& -2& 0\end{array}\right]$.

Theorem: If $S=\left\{\begin{array}{c}\overrightarrow{v_1},\cdots ,\overrightarrow{v_p}\end{array}\right\}$ is an linear independent vectors in ${\mathbb{R}}^n$, then every vector in Span$\left\{\begin{array}{c}\overrightarrow{v_1},\cdots ,\overrightarrow{v_p}\end{array}\right\}$ has a unique representation as a linear combination of $\overrightarrow{v_i}$.

Note: Geometrically, any two vectors in ${\mathbb{R}}^n$ with $n>1$ that are not multiple of each other span a plane( they are not co-line). Any three vectors is an linearly independent set if they are not co-plane.

Theorem: If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set $S=\left\{\begin{array}{c}\overrightarrow{v_1},\cdots ,\overrightarrow{v_p}\end{array}\right\}$ in ${\mathbb{R}}^n$ is linearly dependent if $p>n$.

Theorem: If $S=\left\{\begin{array}{c}\overrightarrow{v_1},\cdots ,\overrightarrow{v_p}\end{array}\right\}$ in ${\mathbb{R}}^n$ contains the zero vector then it is linearly dependent.

Example 3: Use inspection to decide if the vector set is linear independent. State the reasoning.

(a) $\left[\begin{array}{c}2\\ 1\\ -1\end{array}\right]$, $\left[\begin{array}{c}-4\\ -2\\ 2\end{array}\right]$.

(b) $\left[\begin{array}{c}2\\ 1\\ -1\end{array}\right]$, $\left[\begin{array}{c}3\\ 4\\ 5\end{array}\right]$, $\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$.

(c) $\left[\begin{array}{c}2\\ 1\\ -1\end{array}\right]$, $\left[\begin{array}{c}3\\ 4\\ 5\end{array}\right]$, $\left[\begin{array}{c}-2\\ 6\\ -7\end{array}\right]$, $\left[\begin{array}{c}0\\ 2\\ 1\end{array}\right]$.

Exercise 3: Use inspection to decide if the vector set is linear independent. State the reasoning.

(a) $\left[\begin{array}{c}2\\ 1\end{array}\right]$, $\left[\begin{array}{c}-4\\ -2\end{array}\right]$, $\left[\begin{array}{c}4\\ 3\end{array}\right]$.

(b) $\left[\begin{array}{c}2\\ 1\\ 0\\ -1\end{array}\right]$, $\left[\begin{array}{c}3\\ 2\\ 1\\ 5\end{array}\right]$, $\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right]$, $\left[\begin{array}{c}2\\ 1\\ -1\\ 3\end{array}\right]$.

(c) $\left[\begin{array}{c}2\\ 1\\ -1\end{array}\right]$, $\left[\begin{array}{c}3\\ -6\\ 9\end{array}\right]$, $\left[\begin{array}{c}-2\\ 4\\ -6\end{array}\right]$.

Theorem: The following are equivalent for an $n\times n$ matrix $A$:

1. $A$ is invertible.

2. The columns of $A$ are linearly independent.

3. The columns of $A$ span ${\mathbb{R}}^n$.

4. The rows of $A$ are linearly independent.

5. The rows of $A$ span the set of all $n\times n$ rows.

Example 4: Find the value of $h$ such that the columns of $A=\left[\begin{array}{ccc}1& -3& 2\\ 1& 2& h\\ -5& -5& 6\end{array}\right]$ is linearly dependent.

Exercise 4: Find the value of $h$ such that the columns of $A=\left[\begin{array}{ccc}2& -3& h\\ 1& -2& 2\\ -5& 1& 6\end{array}\right]$ is linearly dependent.

Definition: A basis for a subspace $H$ of ${\mathbb{R}}^n$ is a linearly independent set in $H$ that spans $H$.

Fact: The columns of an invertible $n\times n$ matrix form a basis of ${\mathbb{R}}^n$ because they are linearly independent and span ${\mathbb{R}}^n$.

Definition: The columns of $n\times n$ identity has columns $\overrightarrow{e_1}=\left[\begin{array}{c}1\\ 0\\ 0\\ ⋮\\ 0\end{array}\right],\overrightarrow{e_2}=\left[\begin{array}{c}0\\ 1\\ 0\\ ⋮\\ 0\end{array}\right],\cdots ,\overrightarrow{e_n}=\left[\begin{array}{c}0\\ 0\\ 0\\ ⋮\\ 1\end{array}\right]$ which forms a basis of ${\mathbb{R}}^n$. The set $\left\{\begin{array}{c}\overrightarrow{e_n},\cdots ,\overrightarrow{e_n}\end{array}\right\}$ is called standard basis of ${\mathbb{R}}^n$.

Theorem: The pivot columns of a matrix $A$ form a basis for the column space of $A$.

Definition: The dimension of a nonzero subspace $H$, denoted by dim$H$, is the number of vectors in any basis for $H$. The dimension of the zero subspace $\left\{\begin{array}{c}\overrightarrow{0}\end{array}\right\}$ is defined to be zero.

The Basis Theorem: Let $H$ be a $p$-dimensional subspace of ${\mathbb{R}}^n$, any linearly independent set of exactly $p$ elements in $H$ is automatically a basis for $H$. Also, any set of $p$ elements of $H$ that spans $H$ is automatically a basis for $H$.

Example 5: Find a basis and calculate the dimension of the following
subspaces of ${\mathbb{R}}^n$

$U=\{\left[\begin{array}{c}a\\ a+b\\ a-c\\ b\end{array}\right]|a,b,c\text{ in }\mathbb{R}\}$

$V=\{\left[\begin{array}{c}a\\ b\\ c\\ d\end{array}\right]|a+b-c+2d=0\text{ in }\mathbb{R}\}$

Exercise 5: Find a basis and calculate the dimension of the following
subspaces of ${\mathbb{R}}^n$

$U=\{\left[\begin{array}{c}a\\ a+b\\ a-2c\\ c\end{array}\right]|a,b,c\text{ in }\mathbb{R}\}$

$V=\{\left[\begin{array}{c}a\\ b\\ c\\ d\end{array}\right]|a-2b+c+d=0\text{ in }\mathbb{R}\}$

Group Work 1: Mark each statement True or False. Justify each answer.

a. If $B$ is an echelon form of a matrix $A$, then the pivot columns of $B$ form a basis for Col$A$.

b. Row operations do not affect linear dependence relations among the
columns of a matrix.

c. The columns of a matrix $A$ are linearly independent if the equation $A\overrightarrow{x}=\overrightarrow{0}$ has trivial solution.

d. The columns of any $4\times 5$ matrix are linearly dependent.

e. If $\overrightarrow{u}$ and $\overrightarrow{v}$ are linearly independent and if $\overrightarrow{w}$ is in Span$\{\overrightarrow{u},\overrightarrow{v}\}$ then $\{\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}\}$ is linearly dependent.

f. If three vectors in ${\mathbb{R}}^3$ lie on the same plane then they are linearly dependent.

g. If a set contains fewer vectors then there are entries in the vectors then
they are linearly independent.

h. If a set in ${\mathbb{R}}^n$ is linearly dependent then it contains more than $n$ vectors.

Group Work 2: Describe the possible echelon form of the matrix.

(a) $A$ is a $2\times 2$ matrix with linearly independent columns.

(b) $A$ is a $4\times 2$ matrix such that the first column is the multiple of the second column.

Group Work 3: n each case show that the statement is true or give an example showing that it is false.

a. If $\{\overrightarrow{u},\overrightarrow{v}\}$ is independent, then $\{\overrightarrow{u},\overrightarrow{v},\overrightarrow{u}+\overrightarrow{v}\}$ is independent.

b. If $\{\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}\}$ is independent, then $\{\overrightarrow{u},\overrightarrow{v}\}$ is independent.

c. If $\{\overrightarrow{u},\overrightarrow{v}\}$ is dependent, then $\{\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}\}$ is dependent for any $\overrightarrow{w}$.

Group Work 4: How many pivot columns must be a $6\times 4$ matrix have if its columns are linearly independent.

Group Work 5: How many pivot columns must be a $4\times 6$ matrix have if its columns span ${\mathbb{R}}^4$? why?