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Linearly independent vectors

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Wize University Linear Algebra Textbook > Vectors

Vector Operations and Linear Combinations

7 Activities

Wize University Linear Algebra Textbook > Vector Spaces and Subspaces

Linear Independence

4 Activities

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Is this set of vectors {[1 3], [2,-2]} linearly independent? Write [4,-2] as a linear combination of this set.



More Vector Operations and Linear Combinations Questions:
Practice Question
Is the vector (π,e,1)(\pi, e, 1) in the linear span of (1,2,3),(−1,0,1)(1, 2, 3), (-1, 0, 1) and (5,1,−2)(5, 1, -2) of R3\mathbb{R}^3 ? Demonstrate a linear combination that does this.
Find the value of h so that (1,-6,4,h) is a linear combination of (1,-2,1,-4), (0,-4,-1,3) and (0,0,2,5).
Practice: Linear Combination
Show that if u⃗∈Rn\vec u \in \mathbb{R}^n is a linear combination of x⃗,y⃗∈Rn\vec x,\vec y \in \mathbb{R}^n and v⃗∈Rn\vec v \in \mathbb{R}^n is a (different) linear combination of the same x⃗,y⃗\vec x,\vec y then every linear combination au⃗+bv⃗a\vec u+b\vec v for a,b∈Ra,b\in\mathbb{R} is also a linear combination of x⃗,y⃗\vec x,\vec y .
133 - FML 3 - 18.1W e.g. 23
Give a mathematical expression that mean v⃗\bcb{\vec{v}} and w⃗\bcb{\vec{w}} are parallel.
Practice: Vector Operations
Practice: Vector Operations
If u⃗=[1, 0, −1]\vec{u}=\left[1,\ 0,\ -1\right], v⃗=[−2, 4, 1]\vec{v}=\left[-2,\ 4,\ 1\right], and w⃗=[3, 9, −12]\vec{w}=\left[3,\ 9,\ -12\right], determine 5u⃗−12v⃗+23w⃗5\vec{u}-\frac{1}{2}\vec{v}+\frac{2}{3}\vec{w}.
Vector Operations
Practice: Collinear Vectors
Find the value of cc such that the vectors u⃗=⟨−3,c,3⟩\vec u = \lang −3,c,3 \rang and v⃗=⟨1,−2,−1⟩\vec v = \lang 1,−2,−1 \rang are collinear.
Practice: Vector Operations
Practice: Vector Operations
If u⃗=[1, 0, −1]\vec{u}=\left[1,\ 0,\ -1\right], v⃗=[−2, 4, 1]\vec{v}=\left[-2,\ 4,\ 1\right], and w⃗=[3, 9, −12]\vec{w}=\left[3,\ 9,\ -12\right], determine 5u⃗−12v⃗+23w⃗5\vec{u}-\frac{1}{2}\vec{v}+\frac{2}{3}\vec{w}.
Linear combinations
If u⃗=[5−23]\vec{u}=\begin{bmatrix}5\\-2\\3\end{bmatrix}is a linear combination of v⃗=[−2k3]\vec{v}=\begin{bmatrix}-2\\k\\3\end{bmatrix}and w⃗=[32−1]\vec{w}=\begin{bmatrix}3\\2\\-1\end{bmatrix}, what is the value of kk?
Write 1 Vector as linear combination of others
Write (16,−15,8)(16,-15, 8) as a linear combination of (1,1,1),(1,1,0),(1,0,−1)(1,1,1), (1,1,0),(1,0,-1), or show that it is not possible to do so.
133 - FML 3 - 18.1W e.g. 49
Given v⃗=<0, 4, 3>\bcb{\ol{v} = \left< 0,\, 4,\, 3 \right>}, find a unit vector perpendicular to v⃗\bcb{\vec{v}}.
133 - FML 3 - 18.1W e.g. 42
Find a vector parallel to u⃗=<1, −1, 1>\bcb{\vec{u} = \left< 1,\, -1,\, 1\right>}.
133 - FML 3 - 18.1W e.g. 45
Given the points P=(3, 4, −1)\bcb{P = (3,\, 4,\, -1)} and Q=(−1, 2, 2)\bcb{Q = (-1,\, 2,\, 2)}, and the vector u⃗=PQ⃗\bcb{\ol{u} = \vec{PQ}}, find a vector parallel but in the opposite direction to u⃗\bcb{\vec{u}}.
19.4F_Final_Builder_Ch_4.6_Span_Line_From_FAQ_$\tkco{eg}$_$\key{Final}$_Builder_$\tkcth{4.6.}\tkcf{6}$_$\key{|Strc>5.2}$
Describe the set of all linear combinations (sums of constant multiples of) the vectors:
v⃗=[111]\vv = \colvecth{1}{1}{1} and w⃗=[−121]\vw = \colvecth{-1}{2}{1}.
Practice Question: When is 1 Vector a Linear Combination of Others (Has a Solution)
Write (6,5,1)(6,5,1) as a linear combination of (−2,−1,1),(2,9,15),(2,3,3)(-2,-1,1), (2,9,15), (2,3,3) , or show that it is not possible to do so.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 49
Given v⃗=<0, 4, 3>\bcb{\ol{v} = \left< 0,\, 4,\, 3 \right>}, find a unit vector perpendicular to v⃗\bcb{\vec{v}}.
Practice Question: When is 1 Vector a Linear Combination of Others
Is it possible to write down u⃗=(1,2,3)\vec{u}=(1,2,3) as a linear combination of v1⃗=(1,0,2),v2⃗=(0,1,2) and v3⃗=(2,1,6)\vec{v_1}=(1,0,2),\vec{v_2}=(0,1,2)\text{ and }\vec{v_3}=(2,1,6)? If yes write down u⃗\vec{u}as a linear combination of other vectors.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 45
Given the points P=(3, 4, −1)\bcb{P = (3,\, 4,\, -1)} and Q=(−1, 2, 2)\bcb{Q = (-1,\, 2,\, 2)}, and the vector u⃗=PQ⃗\bcb{\ol{u} = \vec{PQ}}, find a vector parallel but in the opposite direction to u⃗\bcb{\vec{u}}.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 42
Find a vector parallel to u⃗=<1, −1, 1>\bcb{\vec{u} = \left< 1,\, -1,\, 1\right>}.
Find the value of h so that (1,-6,4,h) is a linear combination of (1,-2,1,-4), (0,-4,-1,3) and (0,0,2,5).
19.4F_Final_Builder_Ch_4.6_Span_Line_From_FAQ_$\tkco{eg}$_$\key{Final}$_Builder_$\tkcth{4.6.}\tkcf{4}$_$\key{|Strc>5.2}$
Describe the set of all constant multiples of the vector:
v⃗=[−121]\vv = \colvecth{-1}{2}{1}.
19.4F_Final_Builder_Ch_4.6_Span_Line_From_FAQ_$\tkco{eg}$_$\key{Final}$_Builder_$\tkcth{4.6.}\tkcf{2}$_$\key{|Strc>5.2}$
Describe the set of all constant multiples of the vector v⃗=[11]\vv = \colvec{1}{1}.
Practice: Properties of Vector Operations
Practice: Properties of Vector Operations
If u⃗, v⃗, w⃗ ∈R3\vec{u},\ \vec{v},\ \vec{w}\ \in R^3, which of the following statements is/are always true?
i) ∣∣v⃗−u⃗∣∣=−∣∣u⃗−v⃗∣∣\left|\left|\vec{v}-\vec{u}\right|\right|=-\left|\left|\vec{u}-\vec{v}\right|\right|
Vector Addition and Scalar Multiplication

Vector Operations
Practice: Collinear Vectors
Find the value of cc such that the vectors u⃗=⟨−3,c,3⟩\vec u = \lang −3,c,3 \rang and v⃗=⟨1,−2,−1⟩\vec v = \lang 1,−2,−1 \rang are collinear.
Practice: Vector Operations
Practice: Vector Operations
If u⃗=[1, 0, −1]\vec{u}=\left[1,\ 0,\ -1\right], v⃗=[−2, 4, 1]\vec{v}=\left[-2,\ 4,\ 1\right], and w⃗=[3, 9, −12]\vec{w}=\left[3,\ 9,\ -12\right], determine 5u⃗−12v⃗+23w⃗5\vec{u}-\frac{1}{2}\vec{v}+\frac{2}{3}\vec{w}.
Vector Operations
Practice: Collinear Vectors
Given u⃗=⟨2,−5,6⟩,  v⃗=⟨−4,10,−12⟩,\vec{u}=\lang 2,−5,6\rang ,\ \ \vec{v}=\lang−4,10,−12\rang , and 𝑤⃗=⟨−2,5,−6⟩𝑤⃗=\lang −2,5,−6 \rang, select all statements that are true.
Vector Operations: Linear Combinations
Find all values of kk such that the vectors u=(−1,2,k,0)\bm{u}=(-1, 2,k,0) and v=(k,−6,6,0)\bm{v}=(k,-6,6,0) are collinear.
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.
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?