High School
SAT
ACT
University
MCAT
LSAT
DAT

Linear Independence and Basis

Related Topics

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.
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
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
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}.
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\}
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 ?