133 - FML 3 - 18.1W - e.g. 42

Basis and Dimension

4 Activities

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} }​1310​​,​2−100​​,​021−1​​,​36−3−2​​  a basis for R4\bcb{\mathbb{R}^4}R4?

No, they are not a basis for R4\mathbb{R}^4R4 as a single vector spans the same space as all the given vectors, combined. 

No, they are not a basis for R4\mathbb{R}^4R4 as two of the vectors span the same space as all the given vectors, combined. 

No, they are not a basis for R4\mathbb{R}^4R4 as three of the vectors span the same space as all the given vectors, combined. 

No, they are not a basis for R4\mathbb{R}^4R4 as four of the vectors span the same space as all the given vectors, combined. 

Yes, the given vectors are a basis for R4\mathbb{R}^4R4. 

There is insufficient information. 

None of the other options. 

I don't know

Example: Linear Independence (pre-Matrix)

Show that for x⃗,y⃗∈R2\vec x,\vec y\in \mathbb{R}^2x,y​∈R2 , if Span{x⃗,y⃗}=R2Span\{\vec x,\vec y\}=\mathbb{R}^2Span{x,y​}=R2, then x⃗,y⃗\vec x,\vec yx,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}^4Q:=span{(0,0,0,0),(1,1,0,0),(0,0,1,1),(1,1,1,1)}⊆R4

Linear Independence and Basis

Let v⃗1,v⃗2,v⃗3,v⃗4\vec v_1,\vec v_2,\vec v_3, \vec v_4v1​,v2​,v3​,v4​ be any four vectors in R4\mathbb{R}^4R4
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\}C={v1​−v2​,v2​−v3​,v3​−v4​,v4​−v1​} cannot be a basis for R4\mathbb{R}^4R4

133-WML2_Quiz-19.4F_eg_17

It is known that the set {v⃗1, v⃗2, … , v⃗n}\{\vv_1,\, \vv_2,\, \dots \,,\, \vv_n\}{v1​,v2​,…,vn​} is a basis for Rn\mathbb{R^n}Rn. If w⃗∈\vw \inw∈Rn\mathbb{R^n}Rn, is the set {v⃗1, v⃗2, … , v⃗n, w⃗}\{\vv_1,\, \vv_2,\, \dots \,,\, \vv_n,\, \vw\}{v1​,v2​,…,vn​,w} 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}^2R2?

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}^+\}V={(x,y)∣x∈R,y∈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)(x1​,y1​)+(x2​,y2​)=(x1​+x2​,y1​y2​)
k⋅(x,y)=(kx,yk)k\cdot(x,y)=(kx,y^k)k⋅(x,y)=(kx,yk)

Which of the following is a basis for R2\mathbb{R}^2R2?

Which of the following is a basis for R2\mathbb{R}^2R2?

Consider the vector space M2,2(R)\mathcal{M}_{2, 2} (\mathbb{R})M2,2​(R) . What is the dimension of this vector space?

Dimension of a subspace

In the vector space P2P_2P2​, what is the dimension of the subspace that contains all polynomials of the form p(x)=a+bx+cx2p(x)=a+bx+cx^2p(x)=a+bx+cx2, such that a=b+ca=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\}}{x,y​,z,w} form a basis for R4\bcb{\mathbb{R}^4}R4, show that the set {x⃗+kz⃗,y⃗, z⃗, w⃗}\bcb{\left\{ \vec{x} + k\vec{z}, \vec{y},\, \vec{z},\, \vec{w} \right\}}{x+kz,y​,z,w} is also a basis, for any scalar k\bcb{k}k. 

Verify if the following vectors form a basis for R3\mathbb{R}^3R3:
{[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\}⎩⎨⎧​​112​​,​1−21​​,​2−11​​⎭⎬⎫​

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\}}⎩⎨⎧​​10−1​​,​2−20​​,​0−22​​,​3−41​​⎭⎬⎫​, 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\}B1​=⎩⎨⎧​​−120​​,​3−52​​,​4−73​​⎭⎬⎫​ 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\}B2​=⎩⎨⎧​​1−43​​,​52−2​​,​4−70​​⎭⎬⎫​ be two bases of R3\mathbb{R}^3R3.
If [x⃗]B1=[−48−7][\vec x]_{\small B_1} = 
\begin{bmatrix}
  -4\\ 8\\ -7\\
\end{bmatrix}[x]B1​​=​−48−7​​, 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 \}{e1​,…,en​} be a basis for an nnn - dimensional vector space VVV , and let L:V→WL : V\rightarrow WL:V→W be a linear map, where WWW is also a vector space. What conditions are necessary for {L(e1),…,L(en)}\{ L(e_1), \dots, L(e_n) \}{L(e1​),…,L(en​)} to be a basis for WWW ?

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If vectors u⃗\vuu, v⃗\vvv, x⃗\vxx and y⃗\vec{y}y​ are vectors in R7\mathbb{R}^7R7 such that u⃗=−2v⃗+0x⃗+4y⃗\vu = -2\vv + 0\vx +4\vec{y}u=−2v+0x+4y​, then a basis for span{u⃗, v⃗, x⃗, y⃗}\text{span}\big\{ \vu,\, \vv,\, \vx,\, \vec{y} \big\}span{u,v,x,y​} is {u⃗, v⃗, y⃗}\big\{ \vu,\, \vv,\, \vec{y} \big\}{u,v,y​}.

Example: Linear Independence (pre-Matrix)

Example:
Show that for x⃗,y⃗∈R2\vec x,\vec y\in \mathbb{R}^2x,y​∈R2 , if Span{x⃗,y⃗}=R2Span\{\vec x,\vec y\}=\mathbb{R}^2Span{x,y​}=R2, then x⃗,y⃗\vec x,\vec yx,y​ are linearly independent.

Basis and Dimension

Practice: Finding a Basis
Consider the vector space M2×2M_{2\times2}M2×2​ 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\}D={[ac​bd​]∈M2×2​​a+d=b+c} is a subspace of M2×2M_{2\times2}M2×2​

Finding a basis and the dimension of a subspace

In the real vector space R3\mathbb{R}^3R3 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\}W={(x,y,z)∈R3∣y+3z=0}
Find a basis for WWW, and use your answer to state the dimension of WWW

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) \} T={(1,0,0),(0,−2,2),(1,1,1)}is a basis of R3\mathbb{R}^3R3.

Basis of a Subspace

Find a basis for the following subspace of R3\mathbb{R}^3R3 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)\}\}S:={x∈R3 | x⋅v=0 for every v∈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}\}B={v1​​,v2​​,v3​​} is a basis for a vector space VVV
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}u1​​=v1​​−v2​​u2​​=v2​​−v3​​u3​​=−v3​​

Basis and Dimension

Example: Verifying a Basis
Let VVV be the vector space {(x,y)  ∣  x,y∈R+}\{(x,y)\;|\;x,y\in\mathbb{R}^+\}{(x,y)∣x,y∈R+} with operations:
(x1,y1)+(x2,y2)=(x1x2, y1y2)(x_1,y_1)+(x_2,y_2)=(x_1x_2,\ y_1y_2)(x1​,y1​)+(x2​,y2​)=(x1​x2​, y1​y2​)

Find a basis for a given space

Consider the vector space M2×2M_{2\times2}M2×2​ 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\}D={[ac​bd​]∈M2×2​​a+d=b+c} is a subspace of M2×2M_{2\times2}M2×2​
Find a basis for DDD and the dimension of DDD

Show that a set is a basis

Let A=[1−11]A=\left[\begin{array}{rrr}
1&-1&1\\
\end{array}\right]A=[1​−1​1​] and consider the subspace of R3\mathbb{R}^3R3:
N={x→=(x,y,z)∈R3  ∣  A.x→=0→}N=\{\overrightarrow{x}=(x,y,z)\in\mathbb{R}^3\;|\;A.\overrightarrow{x}=\overrightarrow{0}\}N={x=(x,y,z)∈R3∣A.x=0}
Show that B={(1,1,0),(−1,0,1)}B=\{(1,1,0), (-1,0,1)\}B={(1,1,0),(−1,0,1)} is a basis for NNN

Consider the space of polynomials of degree 3 or lower, P3\mathcal{P}_3P3​ . Give an example of a linearly independent set of vectors whose span is the entirety of P3\mathcal{P}_3P3​ .

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}​110​​,​2c1​​,​c43​​ form a basis for R3\mathbb{R}^3R3?

Which of the following sets are a basis for R3\mathbb{R}^3R3 ?

133 - FML 3 - 18.1W - e.g. 42

Related Topics

Basis and Dimension

Are the vectors $\bcb{\begin{bmatrix} 1 \\ 3 \\ 1 \\ 0 \end{bmatrix}, \, \begin{bmatrix} 2 \\ -1 \\ 0 \\ 0 \end{bmatrix}, \, \begin{bmatrix} 0 \\ 2 \\ 1 \\ -1 \end{bmatrix} , \, \begin{bmatrix} 3 \\ 6 \\ -3 \\ -2 \end{bmatrix} }$ a basis for $\bcb{\mathbb{R}^4}$ ?

More Basis and Dimension Questions:

Example: Linear Independence (pre-Matrix)

Practice Question: Bases

Linear Independence and Basis

133-WML2_Quiz-19.4F_eg_17

Show that a subset is a subspace

Dimension of a subspace

133 - FML 3 - 18.1W - e.g. 51

133 - FML 3 - 18.1W - e.g. 44

Practice: Coordinates

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Example: Linear Independence (pre-Matrix)

Basis and Dimension

Finding a basis and the dimension of a subspace

Concept Clarifier

Basis of a Subspace

Example: Abstract Basis Proof

Basis and Dimension

Find a basis for a given space

Show that a set is a basis