High School
SAT
ACT
University
MCAT
LSAT
DAT

Linear Independence

Related Topics

Wize University Linear Algebra Textbook > Vector Spaces and Subspaces

Linear Independence

4 Activities

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.

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.
More Linear Independence Questions:
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.
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?