High School
SAT
ACT
University
MCAT
LSAT
DAT

Let (V, +, ⋅) be a real vector space. If v_1, , v_n is a set of vectors that sp…

Related Topics

Wize University Linear Algebra Textbook > Vector Spaces and Subspaces

Span

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 (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

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 .
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}.
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.
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?