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Linear Independence

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Wize University Linear Algebra Textbook > Vector Spaces and Subspaces

Linear Independence

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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.
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 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?