High School
SAT
ACT
University
MCAT
LSAT
DAT

Let Z={ x = (x);|;x>3} with the following operations: u+ v=uv-3(u+v)+12 k⋅ v …

Related Topics

Wize University Linear Algebra Textbook > Vector Spaces and Subspaces

Vector Spaces

5 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 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
Find the zero vector in this 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)
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
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)