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19.4F_Final_Builder_Ch_4.6_Span_Line_From_FAQ_$\tkco{eg}$_$\key{Final}$_Builder…

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Wize University Linear Algebra Textbook > Vector Spaces and Subspaces

Span

4 Activities

Wize University Linear Algebra Textbook > Vectors

Vector Operations and Linear Combinations

7 Activities

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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}.



More Span Questions:
Practice: Span
Show that for all w⃗,x⃗,y⃗,z⃗∈Rn\vec w,\vec x,\vec y, \vec z\in\mathbb{R}^n , if w⃗∈Span{x⃗,y⃗,z⃗}\vec w\in Span\{\vec x, \vec y, \vec z\} then w⃗∈Span{x⃗−y⃗,y⃗−z⃗,z⃗}\vec w\in Span\{\vec x-\vec y, \vec y-\vec z,\vec z\} .
When is a Vector in the Span of Others
Determine if x⃗=(16,−15,8)∈Span{u⃗,v⃗,w⃗}\vec x=(16,-15, 8) \in \text{Span}\{\vec u, \vec v, \vec w\} if:
u⃗=(1,1,1),v⃗=(1,1,0),w⃗=(1,0,−1)\vec u = (1,1,1), \vec v = (1,1,0), \vec w =(1,0,-1)
Span
Practice: Span
Let u⃗=⟨1,0,1⟩, v⃗=⟨1,1,1⟩, w⃗=⟨1,−1,1⟩\vec u = \lang 1,0,1\rang,\ \vec v = \lang 1,1,1\rang,\ \vec w = \lang 1,-1,1 \rang be vectors in R3\reals^3, and suppose b⃗=⟨3,y,z⟩∈span({u⃗,v⃗,w⃗})\vec b = \lang 3,y,z \rang \in \text{span}(\{\vec u, \vec v, \vec w\}).
Find the value of zz.
Practice: Span
Let u⃗=(2,2,1),v⃗=(2,1,2),w⃗=(2,5,−2),x⃗=(x,y,z)\vec u = (2,2,1), \vec v = (2,1,2), \vec w = (2,5,-2), \vec x = (x,y,z) be vectors in R3\mathbb{R}^3
Find the conditions that the values x,y,zx,y,z must satisfy in order for x⃗\vec x to be in Span({u⃗,v⃗,w⃗})\text{Span}(\{\vec u, \vec v, \vec w\})
Span
Practice: Span
Let u⃗=⟨1,0,1⟩, v⃗=⟨1,1,1⟩, w⃗=⟨1,−1,1⟩\vec u = \lang 1,0,1\rang,\ \vec v = \lang 1,1,1\rang,\ \vec w = \lang 1,-1,1 \rang be vectors in R3\reals^3, and suppose b⃗=⟨3,y,z⟩∈span({u⃗,v⃗,w⃗})\vec b = \lang 3,y,z \rang \in \text{span}(\{\vec u, \vec v, \vec w\}).
Find the value of zz.
Span
Select all of the vectors that are in Span(u⃗,v⃗)\text{Span}(\vec u, \vec v) where u⃗=(−1,1,2),v⃗=(2,1,−3)\vec u=(-1,1,2), \vec v=(2,1,-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)\}?
133 - FML 3 - 18.1W - e.g. 31
Knowing that
[06330−7151−113]→RREF[100010001000]\bcb{\begin{bmatrix}%[c] 0 & 6 & 3 \\ 3 & 0 & -7 \\ 1 & 5 & 1 \\ -1 & 1 & 3 \end{bmatrix} % \xrightarrow{\text{{\tiny RREF}}} % \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}},
determine whether the vector v⃗1=<0,3,1,−1>\bcb{\vec{v}_1 = \big< 0,3,1,-1\big>} is in the span of the vectors v⃗2=<6,0,5,1>\bcb{\vec{v}_2 = \big< 6,0,5,1\big>} and v⃗3=<3,−7,1,3>\bcb{\vec{v}_3 = \big< 3,-7,1,3\big>}.
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. 30
Given the standard basis vectors for R2\bcb{\mathbb{R}^{2}}, viz. ı^\bcb{\ihat} and ȷ^\bcb{\jhat} and the vector u⃗=[02]\bcb{\vec{u} = \begin{bmatrix} 0 \\ 2 \end{bmatrix}}, determine whether ı^\bcb{\ihat} is in the span ⁣{u⃗,ȷ^}\bcb{ \vspan{\vec{u}, \jhat} }.
Briefly justify your answer.
Let (V,+,⋅)(V, +, \cdot) be a real vector space. If v1,…,vnv_1, \dots, v_n is a set of vectors that spans the entirety of VV , prove that the following set of vectors also spans VV .
v1−v2,v2−v3,…,vn−1−vn,vnv_1 - v_2, v_2 - v_3, \dots, v_{n - 1} - v_n, v_n
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 Question: Arbitrary Vector as Linear Combination
Show that every vector in R2\mathbb{R}^2 is a linear combination of (1,−1)(1,-1) and (2,1)(2,1) .
Arbitrary Vector as Linear Combination
Show every vector in R3\mathbb{R}^3 is a combination of (8,0,0),(0,1,1),(0,2,3)(8,0,0),(0,1,1),(0,2,3).
Suppose that S=span([1−12],[−33−6],[121],[547])S=span\left(\begin{bmatrix}1\\-1\\2\end{bmatrix},\begin{bmatrix}-3\\3\\-6\end{bmatrix},\begin{bmatrix}1\\2\\1\end{bmatrix},\begin{bmatrix}5\\4\\7\end{bmatrix}\right) , which of the following statements is true?
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).
Linearly independent vectors
Is this set of vectors {[1 3], [2,-2]} linearly independent? Write [4,-2] as a linear combination of this set.
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}}.
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.