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Verify if the following vectors form a basis for R^3: { 1 1 2 , 1 -2 1 , …

Related Topics

Wize University Linear Algebra Textbook > Vector Spaces and Subspaces

Coordinates

4 Activities

Wize University Linear Algebra Textbook > Vector Spaces and Subspaces

Basis and Dimension

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.
  • Verify if the following vectors form a basis for R3\mathbb{R}^3:
{[112],[1−21],[2−11]}\left\{ \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix},\begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix} \right\}

  • Express the vector [−13−2]\begin{bmatrix} -1\\ 3\\ -2 \end{bmatrix} as a linear combination of elements of this basis.
More Coordinates Questions:
Coordinates
Practice: Coordinates
Find the coordinates of the polynomial p(x)=5+x−3x2−3x3p(x)=5+x-3x^2-3x^3 relative to the following basis of P3P_3:
B={(1+x),  (x+x2),  (x2+x3),  (1−x3)}B = \{(1+x),\ \ (x+x^2),\ \ (x^2+x^3),\ \ (1-x^3)\}
Practice: Coordinates
Find the coordinates of the polynomial p(x)=5+x−3x2−3x3p(x)=5+x-3x^2-3x^3 relative to the following basis of P3P_3:
{(1+x),(x+x2),(x2+x3),(1−x3)}\{(1+x),(x+x^2),(x^2+x^3),(1-x^3)\}
19.4F_Mid_Builder_$\tkcth{8.5.}\tkco{9}$__$\tkco{Mock 3}$
Taken together,
v⃗1=[11]\vv_1 = \colvec{1}{1} and v⃗2=[1−1]\vv_2 = \colvec{1}{-1}
constitute a basis forR2\mathbb{R}^2. Determine the coordinates [x⃗]v=[xv1xv2][ \vx ]_{v} = \colvec{\bcth{x_{v_1}}}{\bct{x_{v_2}}}of the vector x⃗=[21]\vx = \colvec{2}{1} relative to this basis.
Concept Clarifier: Finding Coordinate Vectors
The following is a basis for R2\mathbb{R}^2 .
{(11),(−20)}\left \{ \left ( \begin{array}{c} 1 \\ 1 \end{array} \right ), \left ( \begin{array}{c} -2 \\0 \end{array} \right ) \right \}
If v⃗=(2,3)\vec{v} = (2, 3) is a coordinate vector in this basis, what is v⃗\vec{v} in coordinates with respect to the basis
19.4F_Mid_Builder_$\tkcth{8.4.16}$_$\tkcth{mock 1 ✓}$
Find the coordinates (c1, c2)(c_1,\, c_2) for the point P=(8,11)P = (8,11) in terms of the nonstandard basis vectors e⃗1=<1, 1>\ve{1} = \rowvec{1}{1} and e⃗2=<6, 9>\ve{2} = \rowvec{6}{9}.
Coordinates
Find the coordinates of the polynomial p(x)=2−x+x2p(x)=2-x+x^2 relative to the basis {1+x,1+x2,x+x2}\{1+x,1+x^2,x+x^2\} of P2P_2
19.4F_133_8.2_Mock_F1_$\tkco{eg6}$_$\key{Final}$_Builder_$\tkcth{8.2.}\tkcf{6}$_
The set {[21]}\left\{ \colvec{2}{1} \right\} is a basis for the subspace TT, where T⊆R2T \subseteq \mathbb{R}^2.
Give the position vector r⃗P\vr_{_{P}} (in the standard basis) corresponding to the point in TT whose coordinate is given by PT=(3)P_{T} = (3).
If r⃗=[xy]\vr = \colvec{\bco{x}}{\bct{y}}, then
Coordinates
Practice: Coordinates
Find the coordinates of the polynomial p(x)=5+x−3x2−3x3p(x)=5+x-3x^2-3x^3 relative to the following basis of P3P_3:
B={(1+x),  (x+x2),  (x2+x3),  (1−x3)}B = \{(1+x),\ \ (x+x^2),\ \ (x^2+x^3),\ \ (1-x^3)\}
Coordinates
Practice: Coordinates
Let B1={[−120],[3−52],[4−73]}B_1= \left\{ \begin{bmatrix} -1\\ 2 \\0\\ \end{bmatrix} , \begin{bmatrix} 3\\ -5\\ 2\\ \end{bmatrix} , \begin{bmatrix} 4\\ -7\\ 3\\ \end{bmatrix} \right\} and B2={[2−31],[11−1],[−1−30]}B_2= \left\{ \begin{bmatrix} 2\\ -3 \\1\\ \end{bmatrix} , \begin{bmatrix} 1\\ 1\\ -1\\ \end{bmatrix} , \begin{bmatrix} -1\\ -3\\ 0\\ \end{bmatrix} \right\} be two bases of R3\mathbb{R}^3.
Suppose [x⃗]B1=[−48−7][\vec x]_{\small B_1} = \begin{bmatrix} -4\\ 8\\ -7\\ \end{bmatrix}.
Example: Change of Basis
Example: Change of Basis
Suppose we have 2 ordered bases: A=((1,−1),(−2,1)),B=((1,1),(1,2))A=((1,-1),(-2,1)),B=((1,1),(1,2)) .
Compute the change of basis matrix CA→BC_{A\rightarrow B}, and if[v⃗]A=(−3,−4)[\vec v]_A=(-3,-4), then find [v⃗]B[\vec v]_B .
Practice Question: Change of Basis
Suppose we have an ordered basis B=((1,1,1),(2,0,−1),(−1,1,0))B=((1,1,1),(2,0,-1),(-1,1,0)) . Let S=((1,0,0),(0,1,0),(0,0,1))S=((1,0,0),(0,1,0),(0,0,1)) be the standard ordered basis. Compute the change of basis matrix CS→BC_{S\rightarrow B} , and use it to find (2,−1,2)B(2,-1,2)_B
(i.e. find [v⃗]B[\vec v]_B , if [v⃗]S=(2,−1,2)[\vec v]_S=(2,-1,2) ).
Let {1−x,x,1+x2,x−x3,x4}\{ 1 - x, x, 1 + x^2, x - x^3, x^4 \} be the basis for the vector space of polynomials with degree at most 4. What is the coordinate vector for vv in this basis?
v=4−x2+x3−2x4v = 4 - x^2 + x^3 - 2x^4
Consider the vector v=(1,2,3)v = (1, 2, 3) with respect to the standard unit vector basis. What is the coordinate vector for vv in the following basis:
{(110),(010),(−10−2)}\left \{ \left ( \begin{array}{c} 1 \\ 1\\ 0 \end{array} \right ), \left ( \begin{array}{c} 0 \\1 \\ 0 \end{array} \right ) , \left ( \begin{array}{c} -1 \\ 0 \\ -2 \end{array} \right ) \right \}
More Basis and Dimension Questions:
Example: Linear Independence (pre-Matrix)
Show that for x⃗,y⃗∈R2\vec x,\vec y\in \mathbb{R}^2 , if Span{x⃗,y⃗}=R2Span\{\vec x,\vec y\}=\mathbb{R}^2, then x⃗,y⃗\vec x,\vec y are linearly independent.
Practice Question: Bases
Practice Question: Bases
For each of the following subspaces, find the dimension of the subspace by giving a specific example of a basis:
Q:=span{(0,0,0,0),(1,1,0,0),(0,0,1,1),(1,1,1,1)}⊆R4Q:=span\{(0,0,0,0), (1,1,0,0),(0,0,1,1),(1,1,1,1)\}\subseteq\mathbb{R}^4
Linear Independence and Basis
Let v⃗1,v⃗2,v⃗3,v⃗4\vec v_1,\vec v_2,\vec v_3, \vec v_4 be any four vectors in R4\mathbb{R}^4
Explain why the set C={v⃗1−v⃗2,v⃗2−v⃗3,v⃗3−v⃗4,v⃗4−v⃗1}C=\{\vec v_1-\vec v_2,\vec v_2-\vec v_3,\vec v_3-\vec v_4,\vec v_4-\vec v_1\} cannot be a basis for R4\mathbb{R}^4
133-WML2_Quiz-19.4F_eg_17
It is known that the set {v⃗1, v⃗2, … , v⃗n}\{\vv_1,\, \vv_2,\, \dots \,,\, \vv_n\} is a basis for Rn\mathbb{R^n}. If w⃗∈\vw \inRn\mathbb{R^n}, is the set {v⃗1, v⃗2, … , v⃗n, w⃗}\{\vv_1,\, \vv_2,\, \dots \,,\, \vv_n,\, \vw\} linearly in/dependent? Briefly justify your answer (using mathematical notation is preferable to words, if possible).
Which of the following is a basis for R2\mathbb{R}^2?
Show that a subset is a subspace
Let V={(x,y)  ∣  x∈R,y∈R+}V=\{(x,y)\;|\;x\in\mathbb{R},y\in\mathbb{R}^+\} be the vector space with operations:
(x1,y1)+(x2,y2)=(x1+x2,y1y2)(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1y_2)
k⋅(x,y)=(kx,yk)k\cdot(x,y)=(kx,y^k)
Which of the following is a basis for R2\mathbb{R}^2?
Which of the following is a basis for R2\mathbb{R}^2?
Consider the vector space M2,2(R)\mathcal{M}_{2, 2} (\mathbb{R}) . What is the dimension of this vector space?
Dimension of a subspace
In the vector space P2P_2, what is the dimension of the subspace that contains all polynomials of the form p(x)=a+bx+cx2p(x)=a+bx+cx^2, such that a=b+ca=b+c?
133 - FML 3 - 18.1W - e.g. 51
If it is known that the set of vectors {x⃗, y⃗, z⃗, w⃗}\bcb{\left\{ \vec{x},\, \vec{y},\, \vec{z},\, \vec{w} \right\}} form a basis for R4\bcb{\mathbb{R}^4}, show that the set {x⃗+kz⃗,y⃗, z⃗, w⃗}\bcb{\left\{ \vec{x} + k\vec{z}, \vec{y},\, \vec{z},\, \vec{w} \right\}} is also a basis, for any scalar k\bcb{k}.
133 - FML 3 - 18.1W - e.g. 42
Are the vectors [1310], [2−100], [021−1], [36−3−2]\bcb{\begin{bmatrix} 1 \\ 3 \\ 1 \\ 0 \end{bmatrix}, \, \begin{bmatrix} 2 \\ -1 \\ 0 \\ 0 \end{bmatrix}, \, \begin{bmatrix} 0 \\ 2 \\ 1 \\ -1 \end{bmatrix} , \, \begin{bmatrix} 3 \\ 6 \\ -3 \\ -2 \end{bmatrix} } a basis for R4\bcb{\mathbb{R}^4}?
133 - FML 3 - 18.1W - e.g. 44
Find a basis for the set of vectors {[10−1], [2−20], [0−22], [3−41]}\bcb{\left\{ \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix},\, \begin{bmatrix} 2 \\ -2 \\ 0 \end{bmatrix},\, \begin{bmatrix} 0 \\ -2 \\ 2 \end{bmatrix}, \, \begin{bmatrix} 3 \\ -4 \\ 1\end{bmatrix} \right\}}, and determine the dimension of their span.
Practice: Coordinates
Practice: Coordinates
Let B1={[−120],[3−52],[4−73]}B_1= \left\{ \begin{bmatrix} -1\\ 2 \\0\\ \end{bmatrix} , \begin{bmatrix} 3\\ -5\\ 2\\ \end{bmatrix} , \begin{bmatrix} 4\\ -7\\ 3\\ \end{bmatrix} \right\} and B2={[1−43],[52−2],[4−70]}B_2= \left\{ \begin{bmatrix} 1\\ -4 \\3\\ \end{bmatrix} , \begin{bmatrix} 5\\ 2\\ -2\\ \end{bmatrix} , \begin{bmatrix} 4\\ -7\\ 0\\ \end{bmatrix} \right\} be two bases of R3\mathbb{R}^3.
If [x⃗]B1=[−48−7][\vec x]_{\small B_1} = \begin{bmatrix} -4\\ 8\\ -7\\ \end{bmatrix}, find:
Which of the following are a basis for the vector space of polynomials with degree at most 3?
Let {e1,…,en}\{ e_1, \dots, e_n \} be a basis for an nn - dimensional vector space VV , and let L:V→WL : V\rightarrow W be a linear map, where WW is also a vector space. What conditions are necessary for {L(e1),…,L(en)}\{ L(e_1), \dots, L(e_n) \} to be a basis for WW ?
$\tkct{Mock F1}$ 19.4F_Mid_Builder_$\tkcth{8.6.}\tkco{x}$_$\key{original \#1.}\tkct{b}$__$\tkcth{Mock F}\tkct{?}$_$\tkct{no vid}$
If vectors u⃗\vu, v⃗\vv, x⃗\vx and y⃗\vec{y} are vectors in R7\mathbb{R}^7 such that u⃗=−2v⃗+0x⃗+4y⃗\vu = -2\vv + 0\vx +4\vec{y}, then a basis for span{u⃗, v⃗, x⃗, y⃗}\text{span}\big\{ \vu,\, \vv,\, \vx,\, \vec{y} \big\} is {u⃗, v⃗, y⃗}\big\{ \vu,\, \vv,\, \vec{y} \big\}.
Example: Linear Independence (pre-Matrix)
Example:
Show that for x⃗,y⃗∈R2\vec x,\vec y\in \mathbb{R}^2 , if Span{x⃗,y⃗}=R2Span\{\vec x,\vec y\}=\mathbb{R}^2, then x⃗,y⃗\vec x,\vec y are linearly independent.
Basis and Dimension
Practice: Finding a Basis
Consider the vector space M2×2M_{2\times2} with the standard component-wise operations of addition and scalar multiplication
The subset D={[abcd]∈M2×2  ∣  a+d=b+c}D=\left\{ \left[\begin{array}{cc} a&b\\ c&d \end{array}\right]\in M_{2\times2}\;\Big|\;a+d=b+c \right\} is a subspace of M2×2M_{2\times2}
Finding a basis and the dimension of a subspace
In the real vector space R3\mathbb{R}^3 with the standard component-wise operations of vector addition and scalar multiplication, consider the subspace:
W={(x,y,z)∈R3  ∣  y+3z=0}W=\{(x,y,z)\in\mathbb{R}^3\;|\;y+3z=0\}
Find a basis for WW, and use your answer to state the dimension of WW
Concept Clarifier
Determine whether the set T={(1,0,0),(0,−2,2),(1,1,1)}T=\{ (1,0,0),(0,-2,2), (1,1,1) \} is a basis of R3\mathbb{R}^3.
Basis of a Subspace
Find a basis for the following subspace of R3\mathbb{R}^3 and state its dimension:
S:={x⃗∈R3 | x⃗⋅v⃗=0 for every v⃗∈span{(1,−1,1),(1,1,0)}}S:=\{\vec x\in\mathbb{R}^3\text{ | }\vec x\cdot\vec v=0\text{ for every } \vec v\in span\{(1,-1,1),(1,1,0)\}\}
Example: Abstract Basis Proof
Suppose the set B={v1→,v2→,v3→}B=\{\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}\} is a basis for a vector space VV
Define the vectors:
u1→=v1→−v2→u2→=v2→−v3→u3→=−v3→\overrightarrow{u_1}=\overrightarrow{v_1}-\overrightarrow{v_2} \hspace{1cm}\overrightarrow{u_2}=\overrightarrow{v_2}-\overrightarrow{v_3}\hspace{1cm}\overrightarrow{u_3}=-\overrightarrow{v_3}
Basis and Dimension
Example: Verifying a Basis
Let VV be the vector space {(x,y)  ∣  x,y∈R+}\{(x,y)\;|\;x,y\in\mathbb{R}^+\} with operations:
(x1,y1)+(x2,y2)=(x1x2, y1y2)(x_1,y_1)+(x_2,y_2)=(x_1x_2,\ y_1y_2)
Find a basis for a given space
Consider the vector space M2×2M_{2\times2} with the standard component-wise operations of addition and scalar multiplication
The subset D={[abcd]∈M2×2  ∣  a+d=b+c}D=\left\{ \left[\begin{array}{cc} a&b\\ c&d \end{array}\right]\in M_{2\times2}\;\Big|\;a+d=b+c \right\} is a subspace of M2×2M_{2\times2}
Find a basis for DD and the dimension of DD
Show that a set is a basis
Let A=[1−11]A=\left[\begin{array}{rrr} 1&-1&1\\ \end{array}\right] and consider the subspace of R3\mathbb{R}^3:
N={x→=(x,y,z)∈R3  ∣  A.x→=0→}N=\{\overrightarrow{x}=(x,y,z)\in\mathbb{R}^3\;|\;A.\overrightarrow{x}=\overrightarrow{0}\}
Show that B={(1,1,0),(−1,0,1)}B=\{(1,1,0), (-1,0,1)\} is a basis for NN
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 .
For which value(s) of c do [110],[2c1],[c43]\begin{bmatrix}1\\1\\0\end{bmatrix}, \begin{bmatrix}2\\c\\1\end{bmatrix}, \begin{bmatrix}c\\4\\3\end{bmatrix} form a basis for R3\mathbb{R}^3?
Which of the following sets are a basis for R3\mathbb{R}^3 ?