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Consider the set R^2 where the addition operator is defined to be (v_1, w_1) …

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Wize University Linear Algebra Textbook > Vector Spaces and Subspaces

Vector Spaces

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Consider the set R2\mathbb{R}^2 where the addition operator ⊕\oplus is defined to be
(v1,w1)⊕(v2,w2)=(w1+w2,v1+v2)(v_1, w_1) \oplus (v_2, w_2) = (w_1 + w_2, v_1 + v_2)
and scalar multiplication is
α(v,w)=(αv,αw)\alpha(v, w) = (\alpha v, \alpha w)
Is this a real vector space?
More Vector Spaces Questions:
$\tkcth{7.6.}\tkcf{6}$_Ch_7.6_$\tkco{eg\_6}$_133_$\key{F.}$Builder_Vector_Spaces_$\tkct{}$_20.1W
Consider the set S={x}\mathbb{S} = \big\{x\big\} consisting of only one element: xx. Define addition ⊕\oplus and scalar multiplication ⊗\otimes on S\mathbb{S} by:
x⊕x=xandk⊗x=x, x\oplus x = x \quad \textrm{and} \quad k \otimes x = x,
for all k ∈ Rk \, \in \, \mathbb{R}. Determine whether or not S\mathbb{S}, along with these two operations, is a vector space.
$\tkcth{7.6.}\tkcf{14}$_Ch_7.6_$\tkco{eg\_14}$_133_$\key{F.}$Builder_Vector_Spaces_$\tkct{}$_20.1W
Determine whether the set M={[a00b] : a, b ∈ R}\mathbb{M} = \Bigg\{ \sm{a}{0}{0}{b} \, : \, a,\, b \, \in \, \mathbb{R}\Bigg\}, along with the usual matrix addition and scalar multiplication, constitutes a vector space over R\mathbb{R}.
$\tkcth{7.6.}\tkcf{13}$_Ch_7.6_$\tkco{eg\_13}$_133_$\key{F.}$Builder_Vector_Spaces_$\tkct{}$_20.1W
Let S\mathbb{S} be a vector space. Prove that (−k)v⃗=−(kv⃗)=k(−v⃗)=(−1)kv⃗(-k)\vv = -(k\vv) = k(-\vv) = (-1)k\vec{v}, for all k ∈ Rk \, \in \, \mathbb{R} and all v⃗ ∈ S\vv \, \in \, \mathbb{S}.
$\tkcth{7.6.}\tkcf{15}$_Ch_7.6_$\tkco{eg\_15}$_133_$\key{F.}$Builder_Vector_Spaces_$\tkct{}$_20.1W
Let V={(a,b) : a, b ∈ R, b>0}\mathbb{V} = \big\{ (a,b) \, : \, a,\,b \, \in \, \mathbb{R}, \, b > 0 \big\}, and define vector addition by:
(a,b)⊕(c,d)=(ad+bc, bd)(a,b) \oplus (c,d) = (ad+bc,\, bd),
and define scalar multiplication by
Consider the set of all linear maps that consist of rotations of R2\mathbb{R}^2 (recall that a rotation by some angle is a linear map). Is this a vector space, and if so, what is the dimension of this vector space?
$\tkcth{7.6.}\tkcf{10}$_Ch_7.6_$\tkco{eg\_10}$_133_$\key{F.}$Builder_Vector_Spaces_$\tkct{}$_20.1W
Let S\mathbb{S} be a vector space. Prove that for all k ∈ Rk \, \in \, \mathbb{R}, k0⃗=0⃗k\vz = \vz.
$\tkcth{7.6.}\tkcf{12}$_Ch_7.6_$\tkco{eg\_12}$_133_$\key{F.}$Builder_Vector_Spaces_$\tkct{}$_20.1W
Let S\mathbb{S} be a vector space. Prove that (−1)v⃗=−v⃗(-1) \vv = -\vv for all v⃗∈S\vv \in \mathbb{S}.
Let V={(x,y)  ∣  x,y∈R}V=\{(x,y)\;|\;x,y\in\mathbb{R}\} and define the following operations:
(u1,u2)+(v1,v2)=(u1+v1,u2+v2)(u_1,u_2)+(v_1,v_2)=(u_1+v_1,u_2+v_2)
k⋅(v1,v2)=(k2v1,kv2)k\cdot(v_1,v_2)=(k^2v_1,kv_2)
Vector Spaces
Practice: Finding the Additive Identity and Additive Inverse
Let W={⟨x,y,z⟩  ∣  x,y>0,z∈R}W=\{\lang x,y,z \rang\;|\;x,y>0,z\in\mathbb{R}\} with the following operations:
⟨u1,u2,u3⟩+⟨v1,v2,v3⟩=⟨u1v1, u2v2, u3+v3⟩\lang u_1,u_2,u_3 \rang+\lang v_1,v_2,v_3 \rang=\lang u_1v_1,\ u_2v_2,\ u_3+v_3\rang
Let W={(x,y,z)  ∣  x,y>0,z∈R}W=\{(x,y,z)\;|\;x,y>0,z\in\mathbb{R}\} with the following operations:
(u1,u2,u3)+(v1,v2,v3)=(u1v1,u2v2,u3+v3)(u_1,u_2,u_3)+(v_1,v_2,v_3)=(u_1v_1,u_2v_2,u_3+v_3)
k⋅(v1,v2,v3)=(v1k,v2k,kv3)k\cdot(v_1,v_2,v_3)=(v_1^k,v_2^k,kv_3)
Let Z={x⃗=(x)  ∣  x>3} Z=\{\vec x=(x)\;|\;x>3\} with the following operations:
u⃗+v⃗=uv−3(u+v)+12\vec u+\vec v=uv-3(u+v)+12
k⋅v⃗=(v−3)a+3k\cdot \vec v = (v-3)^a+3
$\tkcth{7.6.}\tkcf{3}$_Ch_7.6_$\tkco{eg\_3}$_133_$\key{F.}$Builder_Vector_Spaces_$\tkct{}$_20.1W
Consider the set F[0,1]={f : [0, 1] → R : f is a function }\mathcal{F}[0,1] = \big\{ f \, : \, [ 0,\, 1] \, \to \, \mathbb{R} \, :\, f \textrm{ is a function } \big\}, with addition and scalar multiplication defined by:
f⊕g=(f+g)(x)=f(x)+g(x)andk⊙f(x)=(kf)(x)=k(f(x)) f\oplus g = (f+g)(x) = f(x) + g(x) \quad \textrm{and} \quad k\odot f(x) = (kf)(x) = k\big(f(x)\big)
Determine whether or not F[0, 1]\mathcal{F}[0,\,1] is a vector space.
$\tkcth{7.6.}\tkcf{8}$_Ch_7.6_$\tkco{eg\_8}$_133_$\key{F.}$Builder_Vector_Spaces_$\tkct{}$_20.1W
Let S\mathbb{S} be a vector space. Prove that the zero vector is unique. That is, suppose there are two vectors x⃗\vx and y⃗\vec{y} in S\mathbb{S} with the property x⃗+v⃗=v⃗\vx + \vv = \vv and y⃗+v⃗=v⃗\vec{y}+\vv = \vv for all v⃗ ∈ S\vv \, \in \, \mathbb{S}, then show that this implies x⃗=y⃗\vx = \vec{y}.
$\tkcth{7.6.}\tkcf{9}$_Ch_7.6_$\tkco{eg\_9}$_133_$\key{F.}$Builder_Vector_Spaces_$\tkct{}$_20.1W
Let S\mathbb{S} be a vector space. Prove that every vector has a unique negative.
That is, for v⃗ ∈S\vv \, \in \mathbb{S}, suppose there are two vectors x⃗\vx and y⃗\vec{y} in S\mathbb{S} with the property v⃗+x⃗=0⃗\vv + \vx = \vz and v⃗+y⃗=0⃗\vv + \vec{y} = \vz; show that this implies x⃗=y⃗\vx = \vec{y}.
Let V=R2V=\mathbb{R}^2 but with the defined the operations:
u⃗+v⃗=(u1+v2,u2+v1)\vec u + \vec v = (u_1+v_2,u_2+v_1)
In this space the zero vector is 0⃗=(0,0)\vec 0 = (0,0)\qquad (can you prove that?)
Let V={(u1,u2)  ∣  u1>0}V=\{(u_1,u_2)\;|\;u_1>0\} and define the operation:
u⃗+v⃗=(u1v1,u2+v2)\vec u + \vec v = (u_1v_1,u_2+v_2)
If this set satisfies the vector space axioms, what is the zero vector in this space?
Abstract vector spaces
Let V={x⃗=(x)  ∣  x∈R,x>3}V=\{\vec x=(x)\;\big|\;x\in\mathbb{R},x>3\}, that is VV is the set of all vectors that are real numbers in the interval (3,∞)(3,\infty)
Define the following operations of vector addition and scalar multiplication:
x⃗+y⃗=xy−3(x+y)+12\vec x + \vec y = xy-3(x+y)+12
Practice: Vector Properties
Prove the following vector property:
u⃗+v⃗=v⃗+u⃗\vec{u}+\vec{v}=\vec{v}+\vec{u}
Let W={(x,y,z)  ∣  x,y>0,z∈R}W=\{(x,y,z)\;|\;x,y>0,z\in\mathbb{R}\} with the following operations:
(u1,u2,u3)+(v1,v2,v3)=(u1v1,u2v2,u3+v3)(u_1,u_2,u_3)+(v_1,v_2,v_3)=(u_1v_1,u_2v_2,u_3+v_3)
k⋅(v1,v2,v3)=(v1k,v2k,kv3)k\cdot(v_1,v_2,v_3)=(v_1^k,v_2^k,kv_3)
Let Z={x⃗=(x)  ∣  x>3}Z=\{\vec x = (x)\;|\;x>3\} with the following operations:
u⃗+v⃗=uv−3(u+v)+12\vec u+\vec v=uv-3(u+v)+12
k⋅v⃗=(v−3)a+3k\cdot \vec v = (v-3)^a+3
Vector Space and Subspace

Consider the vector space, C([0,1])\mathcal{C}([0, 1]) , the space of continuous functions on the interval [0,1][0, 1] . Is this vector space finite-dimensional, or infinite-dimensional? Why?
Vector Spaces
Practice: Finding the Additive Identity and Additive Inverse
Let W={⟨x,y,z⟩  ∣  x,y>0,z∈R}W=\{\lang x,y,z \rang\;|\;x,y>0,z\in\mathbb{R}\} with the following operations:
⟨u1,u2,u3⟩+⟨v1,v2,v3⟩=⟨u1v1, u2v2, u3+v3⟩\lang u_1,u_2,u_3 \rang+\lang v_1,v_2,v_3 \rang=\lang u_1v_1,\ u_2v_2,\ u_3+v_3\rang
Vector Spaces
Practice: Verifying Vector Space Axioms
Let V={(x,y)  ∣  x,y∈R}V=\{(x,y)\;|\;x,y\in\mathbb{R}\} and define the following operations:
(u1,u2)+(v1,v2)=(u1+v1+1, u2+v2+1)(u_1,u_2)+(v_1,v_2)=(u_1+v_1+1,\ u_2+v_2+1)