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Show that a subset is a subspace

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Wize University Linear Algebra Textbook > Vector Spaces and Subspaces

Basis and Dimension

4 Activities

Wize University Linear Algebra Textbook > Vector Spaces and Subspaces

Subspaces

3 Activities

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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)
Note that the zero vector in VV is: 0→=(0,1)\overrightarrow{0}=(0,1)

Consider the subset W={(x,y)∈V  ∣  y=ex}W=\{(x,y)\in V\;|\;y=e^x\}

(a) Show that WW is a subspace of VV

(b) Find a basis for WW and state its dimension
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?
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 ?
More Subspaces Questions:
Span
Practice: Spans are Subspaces
Suppose that v⃗1,…,v⃗k∈Rn\vec v_1, \dots, \vec v_k\in\reals^n. Prove that U=span⁡({v⃗1,...,v⃗k})U=\operatorname{span}(\{\vec v_1, ...,\vec v_k\}) is a subspace of Rn\mathbb{R}^n.
Subspaces
Practice: Subspaces
Let M2×2M_{2\times2} be the vector space whose vectors are 2×22\times 2 matrices with real number entries, and with the standard operations of matrix addition and scalar multiplication.
Consider the following subset of M2×2M_{2\times2}:
19.4F_Mid_Builder_$\tkcth{8.6.}\tkco{x}$_$\key{original \#1.}\tkct{d}$
The vectors [−71000]\begin{bmatrix} -7 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, [−10310]\begin{bmatrix} -1 \\ 0 \\ 3 \\ 1 \\ 0 \end{bmatrix}, [20−501]\begin{bmatrix} 2 \\ 0 \\ -5 \\ 0 \\ 1 \end{bmatrix} are a basis for some subspace of R5\mathbb{R}^5.
Show that U={(abcd)∈M2×2(R)  ∣  a+d=b+c}U=\Big\{ \begin{pmatrix} a&b\\ c&d \end{pmatrix}\in M_{2\times 2}(\mathbb{R})\;|\; a+d=b+c\Big\} is a subspace of M2×2(R)M_{2\times2}(\mathbb{R}) over R\mathbb{R}. Find a basis for UU and find its dimension.
Show that W={p(x)∈P2(R)  ∣  p(−1)=0}W=\Big\{ p(x)\in P_2(\mathbb{R})\;|\; p(-1)=0\Big\} is a subspace of P2(R)P_2(\mathbb{R}) over R\mathbb{R}. Find a basis for WWand find its dimension.
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?
Practice Question: Dimension
For subspaces U,VU,Vof Rn\mathbb{R}^n, prove that dim(U+V)=dim(U)+dim(V)−dim(U∩V)dim(U+V) = dim(U)+dim(V)-dim(U\cap V)
Bases of Subspaces
Suppose that U,VU,V are subspaces of R4\mathbb{R}^4 , with dim(U)=2,dim(V)=3dim(U)=2,dim(V)=3 . Show that there is a nonzero vector in R4\mathbb{R}^4 which is in both UU and VV .
Basis of a Subspace
Find a basis for the following subspace of M2×2M_{2\times2} and state its dimension
Q={[abcd]∈M2×2  ∣  a+b=0,c=2d}Q=\left\{\left[\begin{array}{rr} a&b\\ c&d \end{array}\right]\in M_{2\times2}\;\Big|\; a+b=0, c=2d\right\}
Basis of a Subspace
Find a basis for the following subspace of R3\mathbb{R}^3 and state its dimension
S={x⃗∈R3  ∣   [1−2−1]⋅x⃗=0⃗}S=\left\{\vec x\in\mathbb{R}^3\;\Big|\;\ \left[\begin{array}{rrr} 1&-2&-1\\ \end{array}\right]\cdot\vec x=\vec 0\right\}
Practice Question: Subspaces
For the following subset WW of R4\mathbb{R}^4, select the statements which are true:
W={w⃗∈R4  ∣  Aw⃗=5w⃗}W=\{\vec w\in \mathbb{R}^4\;|\;A\vec w = 5\vec w\} where AA is a 4×44\times 4 matrix
Practice Question: Subspaces
Which of the following subsets are subspaces?
U={(x,y,z)∈R3  ∣  x=2y,z=1}U=\{(x,y,z)\in \mathbb{R}^3\;|\;x=2y,z=1\}
W={w⃗∈R4  ∣  Aw⃗=5w⃗}W=\{\vec w \in \mathbb{R}^4\;|\;A\vec w = 5 \vec w\} where AA is a 4×44\times 4 matrix
Practice Question: Subspaces
For the following subset UU of R3\mathbb{R}^3, select the statements which are true:
U={(x,y,z)∈R3  ∣  x=2y,z=1}U=\{(x,y,z)\in \mathbb{R}^3\;|\;x=2y,z=1\}
Practice Question: Subspaces
For the following subset WW of R3\mathbb{R}^3, select the statements which are true:
W={(a,b,c)∈R3  ∣  c<0}W=\{(a,b,c)\in \mathbb{R}^3\;|\;c<0\}
Give an example of a non-empty subset of R2\mathbb{R}^2 that is closed under scalar multiplication, but is not a subspace.
Let VV be a real vector space. Given two subspaces, UU and WW , is their union ever also a subspace? If so, what is the condition required for their union to be a subspace?