Examples from Subspaces All examples from Subspaces.
[ EXAMPLE ] For a fixed non-zero direction vector w ∈ R n w\in\mathbb R^n w ∈ R n
S = { x ∈ R n : x = λ w , λ ∈ R } . S=\{x\in\mathbb R^n:x=\lambda w,\ \lambda\in\mathbb R\}. S = { x ∈ R n : x = λ w , λ ∈ R } . The zero vector is in S S S λ = 0 \lambda=0 λ = 0 x 1 = λ 1 w x_1=\lambda_1w x 1 = λ 1 w x 2 = λ 2 w x_2=\lambda_2w x 2 = λ 2 w
x 1 + x 2 = ( λ 1 + λ 2 ) w ∈ S . x_1+x_2=(\lambda_1+\lambda_2)w\in S. x 1 + x 2 = ( λ 1 + λ 2 ) w ∈ S . For μ ∈ R \mu\in\mathbb R μ ∈ R
μ x 1 = ( μ λ 1 ) w ∈ S . \mu x_1=(\mu\lambda_1)w\in S. μ x 1 = ( μ λ 1 ) w ∈ S . The subspace test shows that S S S R n \mathbb R^n R n
[RIGHT CLICK] [ASK AI] [ EXAMPLE ] Consider
S = { p ∈ P 3 ( R ) : p ( 1 ) = 0 } . S=\{p\in P_3(\mathbb R):p(1)=0\}. S = { p ∈ P 3 ( R ) : p ( 1 ) = 0 } . The zero polynomial belongs to S S S p 1 , p 2 ∈ S p_1,p_2\in S p 1 , p 2 ∈ S
( p 1 + p 2 ) ( 1 ) = p 1 ( 1 ) + p 2 ( 1 ) = 0 , (p_1+p_2)(1)=p_1(1)+p_2(1)=0, ( p 1 + p 2 ) ( 1 ) = p 1 ( 1 ) + p 2 ( 1 ) = 0 , and if p ∈ S p\in S p ∈ S λ ∈ R \lambda\in\mathbb R λ ∈ R
( λ p ) ( 1 ) = λ p ( 1 ) = 0. (\lambda p)(1)=\lambda p(1)=0. ( λ p ) ( 1 ) = λ p ( 1 ) = 0. Thus S S S P 3 ( R ) P_3(\mathbb R) P 3 ( R ) { 0 } \{0\} { 0 } x − 1 ∈ S x-1\in S x − 1 ∈ S
[RIGHT CLICK] [ASK AI] [ EXAMPLE ] For a fixed x ∈ R n x\in\mathbb R^n x ∈ R n
S = { A ∈ M m × n ( R ) : A x = 0 } . S=\{A\in M_{m\times n}(\mathbb R):Ax=0\}. S = { A ∈ M m × n ( R ) : A x = 0 } . The zero matrix belongs to S S S A 1 , A 2 ∈ S A_1,A_2\in S A 1 , A 2 ∈ S
( A 1 + A 2 ) x = A 1 x + A 2 x = 0. (A_1+A_2)x=A_1x+A_2x=0. ( A 1 + A 2 ) x = A 1 x + A 2 x = 0. If A ∈ S A\in S A ∈ S λ ∈ R \lambda\in\mathbb R λ ∈ R
( λ A ) x = λ ( A x ) = 0. (\lambda A)x=\lambda(Ax)=0. ( λ A ) x = λ ( A x ) = 0. Therefore S S S M m × n ( R ) M_{m\times n}(\mathbb R) M m × n ( R )
[RIGHT CLICK] [ASK AI] [ EXAMPLE ] For a fixed non-zero u ∈ R n u\in\mathbb R^n u ∈ R n
{ x ∈ R n : x = λ u , − 10 ≤ λ ≤ 10 } \{x\in\mathbb R^n:x=\lambda u,\ -10\leq\lambda\leq10\} { x ∈ R n : x = λ u , − 10 ≤ λ ≤ 10 } is not closed under addition: 10 u 10u 10 u 10 u + 10 u = 20 u 10u+10u=20u 10 u + 10 u = 20 u
The affine line
{ x ∈ R 2 : x 1 + 2 x 2 = 1 } \{x\in\mathbb R^2:x_1+2x_2=1\} { x ∈ R 2 : x 1 + 2 x 2 = 1 } does not contain the zero vector. Likewise, neither
{ p ∈ P 3 ( R ) : p ( x ) = 1 for every x ∈ R } nor { p ∈ P 2 ( R ) : p ( 0 ) = 1 } \{p\in P_3(\mathbb R):p(x)=1\text{ for every }x\in\mathbb R\}
\qquad\text{nor}\qquad
\{p\in P_2(\mathbb R):p(0)=1\} { p ∈ P 3 ( R ) : p ( x ) = 1 for every x ∈ R } nor { p ∈ P 2 ( R ) : p ( 0 ) = 1 } contains the zero polynomial. Finally, the invertible matrices in
M n × n ( R ) M_{n\times n}(\mathbb R) M n × n ( R )
Replacing the right-hand side 1 1 1 0 0 0 p ( x ) = 0 p(x)=0 p ( x ) = 0 x x x p ( 0 ) = 0 p(0)=0 p ( 0 ) = 0 linear
transformations .
[RIGHT CLICK] [ASK AI] Subspaces Builds on Vector spaces .