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Subspaces

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Wize University Linear Algebra Textbook > Vector Spaces and Subspaces

Subspaces

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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}:
W={[abcd]∈M2×2 ∣ a+d=b+c}W=\left\{\left[\begin{array}{rr} a&b\\ c&d\\ \end{array}\right]\in M_{2\times2}\ \Big |\ a+d=b+c\right\}
Show that WW is a subspace of M2×2M_{2\times2}.
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.
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)
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?