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Wize University Linear Algebra Textbook > Vector Spaces and Subspaces

Linear Independence

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Linear Independence

Definition

Suppose you have a set of vectors {v⃗1,v⃗2,…,v⃗n}\{\vec v_1,\vec v_2,\dots,\vec v_n\}, and let c1,c2,…,cn∈Rc_1, c_2, \dots, c_n \in \reals.
Consider the equation: c1v⃗1+c2v⃗2+⋯+cnv⃗n=0⃗c_1\vec v_1 + c_2\vec v_2 + \dots + c_n\vec v_n = \vec 0.
If the only solution to this equation is c1=c2=…=cn=0c_1 = c_2 = \ldots = c_n = 0, then the set {v⃗1,v⃗2,…,v⃗n}\{\vec v_1,\vec v_2,\dots,\vec v_n\} is said to be linearly independent.
If there are any non-zero values cic_i that satisfy the equation, the set is linearly dependent.
Wize Tip
If any vector can be written as a linear combination of other vectors, then the set is linearly dependent.

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Testing Linear Independence

Asking if the set of vectors {v⃗1,v⃗2,…,v⃗n}\{\vec v_1,\vec v_2,\dots,\vec v_n\} is linearly independent is equivalent to solving a homogeneous linear system.
Taking the unknowns to be the coefficients c1,c2,…,cnc_1,c_2,\dots,c_n, we can write the augmented matrix of the system:

[∣  ∣  ∣  ∣v⃗1v⃗2⋯v⃗n0⃗∣  ∣  ∣  ∣]\left[\begin{array}{rrrr|r} |\ \ &|\ \ & & |\ \ & | \\ \vec v_1&\vec v_2&\cdots&\vec v_n& \vec 0\\ |\ \ &|\ \ & & |\ \ & | \\ \end{array}\right]
The type of solution determines linear independence:
  • Only the trivial solution (c1=c2=⋯=cn=0c_1=c_2=\cdots=c_n=0)   ⟹  \implies linearly independent
  • Non-trivial solutions   ⟹  \implies linearly dependent
Shortcut: Determinant
The determinant can be used to determine the number of solutions to a homogeneous linear system.

Let A=[v⃗1v⃗2⋯v⃗n]A=\left[\begin{array}{rrrr} \\ \vec v_1&\vec v_2&\cdots&\vec v_n\\ \\ \end{array}\right].
  • det(A)≠0   ⟹  \boxed{\text{det}(A)\textcolor{red}{\neq}0 } \ \implies AA is invertible   ⟹  \impliestrivial solution only   ⟹  \implies linearly independent
  • det(A)=0   ⟹  \boxed{\text{det}(A)=0} \ \implies AA is not invertible   ⟹  \implies infinitely many solutions   ⟹  \implies linearly dependent
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Examples

Linearly Independent
In R2\mathbb{R}^2, the set {⟨1,1⟩,⟨1,2⟩}\{\lang 1,1\rang , \lang 1,2 \rang\} is linearly independent.
c1[11]+c2[12]=[00]  ⟹  1c1+1c2=01c1+2c2=0\begin{array}{rcl} c_1 \begin{bmatrix} 1\\ 1\\ \end{bmatrix} +c_2 \begin{bmatrix} 1\\ 2\\ \end{bmatrix} = \begin{bmatrix} 0\\ 0\\ \end{bmatrix} &\implies& 1c_1&+&1c_2&=&0\\ &&1c_1&+&2c_2&=&0\end{array}
The only solution to this linear system is c1=c2=0c_1=c_2=0.
Check using the determinant:
∣1112∣=(1)(2)−(1)(1)=1≠0\left| \begin{array}{rrr} 1 & 1\\ 1 & 2\\ \end{array} \right| =(1)(2)-(1)(1)=1\textcolor{red}{\ne}0
Linearly Dependent
In R2\mathbb{R}^2, the set {⟨1,2⟩,⟨3,6⟩}\{\lang 1,2 \rang, \lang 3,6 \rang\} is linearly dependent.
c1[12]+c2[36]=[00]  ⟹  1c1+3c2=02c1+6c2=0\begin{array}{rcl} c_1 \begin{bmatrix} 1\\ 2\\ \end{bmatrix} + c_2 \begin{bmatrix} 3\\ 6\\ \end{bmatrix} = \begin{bmatrix} 0\\ 0\\ \end{bmatrix} &\implies& 1c_1&+&3c_2&=&0\\ &&2c_1&+&6c_2&=&0 \end{array}
This linear system has non-zero solutions, such as c1=−3, c2=1c_1=-3,\ c_2=1.
Moreover, we can see that one vector ⟨3,6⟩\lang 3,6 \rang is a linear combination of the other:⟨3,6⟩=3⋅⟨1,2⟩\lang 3,6 \rang = 3\cdot\lang 1,2 \rang.
Check using the determinant:
∣1326∣=(1)(6)−(3)(2)=0\left| \begin{array}{rrr} 1 & 3\\ 2 & 6\\ \end{array} \right| =(1)(6)-(3)(2)=0
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Example: Linear Independence

Let u⃗=[40−2]\vec u = \begin{bmatrix} 4\\ 0\\ -2\\ \end{bmatrix} , v⃗=[137]\vec v= \begin{bmatrix} 1\\ 3\\ 7\\ \end{bmatrix}, and w⃗=[335]\vec w = \begin{bmatrix} 3\\ 3\\ 5\\ \end{bmatrix}.
Determine if the set {u⃗,v⃗,w⃗}\{\vec u, \vec v, \vec w\} is linearly independent.
Using the definition of linear independence, we are looking to find what values of c1,c2c_1, c_2, and c3c_3 are solutions to the equation:
c1u⃗+c2v⃗+c3w⃗=0⃗c_1\vec u + c_2\vec v + c_3\vec w = \vec 0
  ⟹  c1[40−2]+c2[137]+c3[335]=[000]\implies c_1 \begin{bmatrix} 4\\ 0\\ -2\\ \end{bmatrix} + c_2 \begin{bmatrix} 1\\ 3\\ 7\\ \end{bmatrix} + c_3 \begin{bmatrix} 3\\ 3\\ 5\\ \end{bmatrix} = \begin{bmatrix} 0\\ 0\\ 0\\ \end{bmatrix}
  ⟹  {4c1+c2+3c3=03c2+3c3=0−2c1+7c2+5c3=0\implies \left\{\begin{array}{rcl} 4c_1+c_2+3c_3&=&0\\[0.5em] 3c_2+3c_3&=&0\\[0.5em] -2c_1+7c_2+5c_3&=&0 \end{array}\right.
  ⟹  [41300330−2750]\implies \left[\begin{array}{rrr|r}4&1&3&0\\0&3&3&0\\-2&7&5&0\end{array}\right]
The columns of this augmented matrix are the vectors u⃗,v⃗,\vec u, \vec v, and w⃗\vec w.
We can use Gauss-Jordan elimination to find the RREF:[100001000010]\left[\begin{array}{rrr|r}1&0&0&0\\0&1&0&0\\0&0&1&0\end{array}\right]
We can then read of the solution directly: c1=c2=c3=0c_1=c_2=c_3=0.
Instead, let's simply find the determinant of the coefficient matrix:
∣413033−275∣=−0(… ) + 3(4(5)−3(−2)) − 3(4(7)−1(−2))=3(26)−3(30)=−12 ≠ 0\begin{array}{rcl} \left|\begin{array}{rrr}4&1&3\\\colorOne0&\colorOne3&\colorOne3\\-2&7&5\end{array}\right| &=& -\colorOne0(\dots) \ +\ \colorOne3\big(4(5)-3(-2)\big) \ -\ \colorOne3\big(4(7)-1(-2)\big)\\[1em] &=& 3(26)-3(30)\\[1em] &=&-12 \ \textcolor{red}{\neq} \ 0 \end{array}
Since the determinant is not 0, the system has only the trivial solution c1=c2=c3=0c_1=c_2=c_3=0.
Therefore, {u⃗,v⃗,w⃗}\{\vec u, \vec v,\vec w\} is a linearly independent set.
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Example: Linear Independence

Let u⃗=(1,2,1)\vec u = (1,2,1), v⃗=(1,3,2)\vec v=(1,3,2), and w⃗=(7,17,a2+1)\vec w = (7,17,a^2+1).
Determine the value(s) of aa that will make the set {u⃗,v⃗,w⃗}\{\vec u, \vec v, \vec w\} linearly independent.
Let's find the determinant of the matrix whose columns are u⃗,v⃗,\vec u, \vec v, and w⃗\vec w: [117231712a2+1]\left[ \begin{array}{rrc} 1 & 1 & 7\\ 2 & 3 & 17\\ 1 & 2 & a^2+1\\ \end{array} \right]
∣117231712a2+1∣=1(3(a2+1)−17(2))−1(2(a2+1)−17(1))+7(2(2)−3(1))=3a2+3−34−2a2−2+17+7=a2−9\begin{array}{rcl} \left|\begin{array}{ccc} \colorOne1&\colorOne1&\colorOne7\\ 2&3&17\\ 1&2&a^2+1 \end{array}\right| &=& \colorOne1\Big(3(a^2+1)-17(2)\Big) -\colorOne1\Big(2(a^2+1)-17(1)\Big) +\colorOne7\Big(2(2)-3(1)\Big)\\[1em] &=&3a^2+3-34-2a^2-2+17+7\\[1em] &=&a^2-9 \end{array}
Since we want the vectors to be linearly independent, we must make sure the determinant is not 0:
a2−9≠0  ⟹  (a−3)(a+3)≠0  ⟹  a≠3anda≠−3a^2-9\textcolor{red}{\neq}0 \implies (a-3)(a+3)\textcolor{red}{\neq}0 \implies a\textcolor{red}{\neq}3 \quad\text{and} \quad a\textcolor{red}{\neq}-3.
Therefore, as long as aa is not equal to 3 or -3, the vectors will be linearly independent.

Practice: Linear Independence

Let {v⃗1,v⃗2,v⃗3}\{\vec v_1,\vec v_2,\vec v_3\} be a linearly independent set of vectors in a vector space VV.
Define the following vectors in VV, where k∈Rk \in \reals:
w⃗1=2v⃗2+kv⃗3w⃗2=v⃗1+kv⃗2w⃗3=kv⃗1−4v⃗3\vec w_1=2\vec v_2+k\vec v_3\qquad \vec w_2=\vec v_1+k\vec v_2 \qquad \vec w_3=k\vec v_1-4\vec v_3
Determine what conditions kk must satisfy in order for the set {w⃗1,w⃗2,w⃗3}\{\vec w_1,\vec w_2,\vec w_3\} to be linearly independent.
Extra Practice
Linearly independent vectors
Determine variable x such that 𝒖 = (1,1,1) ,𝒗 = (3,2,1) and 𝒘 = (𝑥, 0,1) are linearly dependent.
Linearly independent vectors
Determine variable x such that 𝒖 = (1,1,1) ,𝒗 = (3,2,1) and 𝒘 = (𝑥, 0,1) are linearly dependent.
Determine r and s such that (1,-1,4,1), (-1,1,0,0) and (2,r,s,3) are linearly dependent.
Linearly independent vectors
Is this set of vectors {[1 3], [2,-2]} linearly independent? Write [4,-2] as a linear combination of this set.
Linear Independence
Example: Linear Independence
Let u⃗=[40−2]\vec u = \begin{bmatrix} 4\\ 0\\ -2\\ \end{bmatrix} , v⃗=[137]\vec v= \begin{bmatrix} 1\\ 3\\ 7\\ \end{bmatrix}, and w⃗=[335]\vec w = \begin{bmatrix} 3\\ 3\\ 5\\ \end{bmatrix}.
Determine if the set {u⃗,v⃗,w⃗}\{\vec u, \vec v, \vec w\} is linearly independent.
Practice Question: Linear Independence
Use the definition of linear independence to show that the vectors (1,0,3),(0,−1,0),(2,2,2)∈R3(1,0,3),(0,-1,0),(2,2,2) \in\mathbb{R}^3 are linearly independent.
Practice: Linear Independence
Show that if x⃗+z⃗,x⃗+3⃗z,y⃗+z⃗\vec x+\vec z, \vec x+\vec 3z, \vec y+\vec z are linearly independent vectors in Rn\mathbb{R}^n then x⃗,y⃗,z⃗\vec x,\vec y,\vec z are also linearly independent vectors in Rn\mathbb{R}^n .
Practice Question: Linear Independence
Use the definition of linear independence to show that the vectors (1,0,3),(0,−1,0),(2,2,2)∈R3(1,0,3),(0,-1,0),(2,2,2) \in\mathbb{R}^3 are linearly independent
Linear Independence
Practice: Linear Independence
Let {v⃗1,v⃗2,v⃗3}\{\vec v_1,\vec v_2,\vec v_3\} be a linearly independent set of vectors in a vector space VV.
Define the following vectors in VV, where k∈Rk \in \reals:
Practice: Linear Independence
Let u⃗=(−1,1,0,1)\vec u = (-1,1,0,1), v⃗=(1,0,−1,1)\vec v = (1,0,-1,1), and w⃗=(0,−1,1,−2)\vec w=(0,-1,1,-2)
Determine if the set {u⃗,v⃗,w⃗}\{\vec u, \vec v, \vec w\} is linearly independent
Practice: Linear Independence
Let {v⃗1,v⃗2,v⃗3}\{\vec v_1,\vec v_2,\vec v_3\} be a linearly independent set of vectors in a vector space VV
Now define the following vectors in VV:
w⃗1=2v⃗2+kv⃗3w⃗2=v⃗1+kv⃗2w⃗3=kv⃗1−4v⃗3\vec w_1=2\vec v_2+k\vec v_3\qquad \vec w_2=\vec v_1+k\vec v_2 \qquad \vec w_3=k\vec v_1-4\vec v_3
Linear Independence
Practice: Linear Independence
Let {v⃗1,v⃗2,v⃗3}\{\vec v_1,\vec v_2,\vec v_3\} be a linearly independent set of vectors in a vector space VV.
Define the following vectors in VV, where k∈Rk \in \reals:
Linear Independence and Span concepts
Which of the following statements are true about the set of vectors {(1,1,−1,0),(−1,3,−1,1),(2,0,−3,4)}\{(1,1,-1,0),(-1,3,-1,1),(2,0,-3,4)\}?
Determine which ones are linearly independent:
Linear Independence
Example: Linear Independence
Let u⃗=(1,2,1)\vec u = (1,2,1), v⃗=(1,3,2)\vec v=(1,3,2), and w⃗=(7,17,a2+1)\vec w = (7,17,a^2+1).
Determine the value(s) of aa that will make the set {u⃗,v⃗,w⃗}\{\vec u, \vec v, \vec w\} linearly independent.
Linear Independence
Let u⃗=[40−2]\vec u = \begin{bmatrix} 4\\ 0\\ -2\\ \end{bmatrix} , v⃗=[137]\vec v= \begin{bmatrix} 1\\ 3\\ 7\\ \end{bmatrix}, and w⃗=[335]\vec w = \begin{bmatrix} 3\\ 3\\ 5\\ \end{bmatrix}.
Determine if the set {u⃗,v⃗,w⃗}\{\vec u, \vec v, \vec w\} is linearly independent.
Consider the space of polynomials of degree 3 or lower, P3\mathcal{P}_3 . Give an example of a linearly independent set of vectors whose span is the entirety of P3\mathcal{P}_3 .
133 - FML 3 - 18.1W - e.g. 32
Show that the vectors v⃗1=(1,2,3,4)\bcb{\vec{v}_1 = (1,2,3,4)}, v⃗2=(0,1,0,−1)\bcb{\vec{v}_2 = (0,1,0,-1)}, and v⃗3=(1,3,3,3)\bcb{\vec{v}_3 = (1,3,3,3)} form a linearly dependent set, then express each vector as a linear combination of the other two.
v⃗1=\vv_1 =
-1
v⃗2\vv_2 +
1
v⃗3\vv_3
v⃗2=\vv_2 =
-1
v⃗1\vv_1 +
1
v⃗3\vv_3
Let our real vector space be (R3,+,⋅)(\mathbb{R}^3, +, \cdot) . Are the following set of vectors linearly independent? If not, write one of the vectors as as linear combination of the others.
(−134),(31−2),(1375),(121)\left ( \begin{array}{c} -1\\3\\4 \end{array} \right ), \left ( \begin{array}{c} 3\\1\\-2 \end{array} \right ), \left ( \begin{array}{c} 13\\7\\5 \end{array} \right ), \left ( \begin{array}{c} 1\\2\\1 \end{array} \right )
Let our real vector space be (R3,+,⋅)(\mathbb{R}^3, +, \cdot) . Are the following set of vectors linearly independent? If not, write one of the vectors as a linear combination of the others.
(123),(011),(−111)\left ( \begin{array}{c} 1\\2\\3 \end{array} \right ), \left ( \begin{array}{c} 0\\1\\1 \end{array} \right ), \left ( \begin{array}{c} -1\\1\\1 \end{array} \right )
133- 05.4F - Final - PartI Question#7
For what values of α are the vectors (α,1,1)\bcb{(\alpha, 1, 1)}, (1,α,α)\bcb{(1, \alpha, \alpha)} and (8,7,α)\bcb{(8, 7, \alpha) }linearly dependent ?
133- 05.4F - Final - PartII Question#3b
Suppose that the vectors {v1,v2,v3}\bcb{ \{v1, v2, v3\} } are linearly independent.
If a1=4v1+2v2−2v3\bcb{a1 = 4v1 + 2v2 − 2v3}, a2=v1+3v2−v3\bcb{a2 = v1 + 3v2 − v3}, and a3=3v1−6v2\bcb{a3 = 3v1 − 6v2}, show that the vectors {a1, a2, a3} are not linearly independent.
133- 05.4F - Final - PartII Question#3a
Suppose that the vectors {v1,v2,v3}\bcb{ \{v1, v2, v3\} } are linearly independent.
If w1=v1−v2+v3\bcb{w1 = v1 − v2 + v3}, w2=v1−v3\bcb{w2 = v1 − v3}, and w3=v1+v2+v3\bcb{w3 = v1 + v2 + v3}, show that the vectors {w1,w2,w3}\bcb{ \{w1, w2, w3\}} are linearly independent.
Determine r and s such that (1,-1,4,1), (-1,1,0,0) and (2,r,s,3) are linearly dependent.
Given that u⃗\vec{u}and v⃗\vec{v}are linearly independent, which of the following sets is also linearly independent?