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Consider the space of polynomials of degree 3 or lower, P_3 . Give an example o…

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Wize University Linear Algebra Textbook > Vector Spaces and Subspaces

Span

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Wize University Linear Algebra Textbook > Vector Spaces and Subspaces

Linear Independence

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Wize University Linear Algebra Textbook > Vector Spaces and Subspaces

Basis and Dimension

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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 .
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>}.
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.
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
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?
More Linear Independence Questions:
Linearly independent vectors
Determine variable x such that 𝒖 = (1,1,1) ,𝒗 = (3,2,1) and 𝒘 = (𝑥, 0,1) are linearly dependent.
Linearly independent vectors
Determine variable x such that 𝒖 = (1,1,1) ,𝒗 = (3,2,1) and 𝒘 = (𝑥, 0,1) are linearly dependent.
Determine r and s such that (1,-1,4,1), (-1,1,0,0) and (2,r,s,3) are linearly dependent.
Linearly independent vectors
Is this set of vectors {[1 3], [2,-2]} linearly independent? Write [4,-2] as a linear combination of this set.
Linear Independence
Example: Linear Independence
Let u⃗=[40−2]\vec u = \begin{bmatrix} 4\\ 0\\ -2\\ \end{bmatrix} , v⃗=[137]\vec v= \begin{bmatrix} 1\\ 3\\ 7\\ \end{bmatrix}, and w⃗=[335]\vec w = \begin{bmatrix} 3\\ 3\\ 5\\ \end{bmatrix}.
Determine if the set {u⃗,v⃗,w⃗}\{\vec u, \vec v, \vec w\} is linearly independent.
Practice Question: Linear Independence
Use the definition of linear independence to show that the vectors (1,0,3),(0,−1,0),(2,2,2)∈R3(1,0,3),(0,-1,0),(2,2,2) \in\mathbb{R}^3 are linearly independent.
Practice: Linear Independence
Show that if x⃗+z⃗,x⃗+3⃗z,y⃗+z⃗\vec x+\vec z, \vec x+\vec 3z, \vec y+\vec z are linearly independent vectors in Rn\mathbb{R}^n then x⃗,y⃗,z⃗\vec x,\vec y,\vec z are also linearly independent vectors in Rn\mathbb{R}^n .
Practice Question: Linear Independence
Use the definition of linear independence to show that the vectors (1,0,3),(0,−1,0),(2,2,2)∈R3(1,0,3),(0,-1,0),(2,2,2) \in\mathbb{R}^3 are linearly independent
Linear Independence
Practice: Linear Independence
Let {v⃗1,v⃗2,v⃗3}\{\vec v_1,\vec v_2,\vec v_3\} be a linearly independent set of vectors in a vector space VV.
Define the following vectors in VV, where k∈Rk \in \reals:
Practice: Linear Independence
Let u⃗=(−1,1,0,1)\vec u = (-1,1,0,1), v⃗=(1,0,−1,1)\vec v = (1,0,-1,1), and w⃗=(0,−1,1,−2)\vec w=(0,-1,1,-2)
Determine if the set {u⃗,v⃗,w⃗}\{\vec u, \vec v, \vec w\} is linearly independent
Practice: Linear Independence
Let {v⃗1,v⃗2,v⃗3}\{\vec v_1,\vec v_2,\vec v_3\} be a linearly independent set of vectors in a vector space VV
Now define the following vectors in VV:
w⃗1=2v⃗2+kv⃗3w⃗2=v⃗1+kv⃗2w⃗3=kv⃗1−4v⃗3\vec w_1=2\vec v_2+k\vec v_3\qquad \vec w_2=\vec v_1+k\vec v_2 \qquad \vec w_3=k\vec v_1-4\vec v_3
Linear Independence
Practice: Linear Independence
Let {v⃗1,v⃗2,v⃗3}\{\vec v_1,\vec v_2,\vec v_3\} be a linearly independent set of vectors in a vector space VV.
Define the following vectors in VV, where k∈Rk \in \reals:
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)\}?
Determine which ones are linearly independent:
Linear Independence
Example: Linear Independence
Let u⃗=(1,2,1)\vec u = (1,2,1), v⃗=(1,3,2)\vec v=(1,3,2), and w⃗=(7,17,a2+1)\vec w = (7,17,a^2+1).
Determine the value(s) of aa that will make the set {u⃗,v⃗,w⃗}\{\vec u, \vec v, \vec w\} linearly independent.
Linear Independence
Let u⃗=[40−2]\vec u = \begin{bmatrix} 4\\ 0\\ -2\\ \end{bmatrix} , v⃗=[137]\vec v= \begin{bmatrix} 1\\ 3\\ 7\\ \end{bmatrix}, and w⃗=[335]\vec w = \begin{bmatrix} 3\\ 3\\ 5\\ \end{bmatrix}.
Determine if the set {u⃗,v⃗,w⃗}\{\vec u, \vec v, \vec w\} is linearly independent.
133 - FML 3 - 18.1W - e.g. 32
Show that the vectors v⃗1=(1,2,3,4)\bcb{\vec{v}_1 = (1,2,3,4)}, v⃗2=(0,1,0,−1)\bcb{\vec{v}_2 = (0,1,0,-1)}, and v⃗3=(1,3,3,3)\bcb{\vec{v}_3 = (1,3,3,3)} form a linearly dependent set, then express each vector as a linear combination of the other two.
v⃗1=\vv_1 =
-1
v⃗2\vv_2 +
1
v⃗3\vv_3
v⃗2=\vv_2 =
-1
v⃗1\vv_1 +
1
v⃗3\vv_3
Let our real vector space be (R3,+,⋅)(\mathbb{R}^3, +, \cdot) . Are the following set of vectors linearly independent? If not, write one of the vectors as as linear combination of the others.
(−134),(31−2),(1375),(121)\left ( \begin{array}{c} -1\\3\\4 \end{array} \right ), \left ( \begin{array}{c} 3\\1\\-2 \end{array} \right ), \left ( \begin{array}{c} 13\\7\\5 \end{array} \right ), \left ( \begin{array}{c} 1\\2\\1 \end{array} \right )
Let our real vector space be (R3,+,⋅)(\mathbb{R}^3, +, \cdot) . Are the following set of vectors linearly independent? If not, write one of the vectors as a linear combination of the others.
(123),(011),(−111)\left ( \begin{array}{c} 1\\2\\3 \end{array} \right ), \left ( \begin{array}{c} 0\\1\\1 \end{array} \right ), \left ( \begin{array}{c} -1\\1\\1 \end{array} \right )
133- 05.4F - Final - PartI Question#7
For what values of α are the vectors (α,1,1)\bcb{(\alpha, 1, 1)}, (1,α,α)\bcb{(1, \alpha, \alpha)} and (8,7,α)\bcb{(8, 7, \alpha) }linearly dependent ?
133- 05.4F - Final - PartII Question#3b
Suppose that the vectors {v1,v2,v3}\bcb{ \{v1, v2, v3\} } are linearly independent.
If a1=4v1+2v2−2v3\bcb{a1 = 4v1 + 2v2 − 2v3}, a2=v1+3v2−v3\bcb{a2 = v1 + 3v2 − v3}, and a3=3v1−6v2\bcb{a3 = 3v1 − 6v2}, show that the vectors {a1, a2, a3} are not linearly independent.
133- 05.4F - Final - PartII Question#3a
Suppose that the vectors {v1,v2,v3}\bcb{ \{v1, v2, v3\} } are linearly independent.
If w1=v1−v2+v3\bcb{w1 = v1 − v2 + v3}, w2=v1−v3\bcb{w2 = v1 − v3}, and w3=v1+v2+v3\bcb{w3 = v1 + v2 + v3}, show that the vectors {w1,w2,w3}\bcb{ \{w1, w2, w3\}} are linearly independent.
Determine r and s such that (1,-1,4,1), (-1,1,0,0) and (2,r,s,3) are linearly dependent.
Given that u⃗\vec{u}and v⃗\vec{v}are linearly independent, which of the following sets is also linearly independent?
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}.
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
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 ?