Undergrad Subspaces in Axler's Famous Textbook

[gibberingmouther]

Just started working through "Linear Algebra Done Right". There is something I don't understand.

Given b ∈ F, then
{(x₁,x₂,x₃,x₄) ∈ F⁴ : x₃ = 5x₄ + b}
is a subspace of F⁴ *if and only if* b=0

I just flat out don't understand why b has to be 0 or even what is the point of bringing this up.

and right below that is:
{p ∈ P(F) : p(3) = 0}
is a subspace of P(F).

P(F) refers to the polynomial space. F is the set of fields and it contains C (complex numbers) and R (real numbers).

Again, what is the point of bringing this up and how do we know that p is a subspace of P(F) based off of the information given?

 

[fresh_42]

Just started working through "Linear Algebra Done Right". There is something I don't understand.

Given b ∈ F, then
{(x₁,x₂,x₃,x₄) ∈ F⁴ : x₃ = 5x₄ + b}
is a subspace of F⁴ *if and only if* b=0

I just flat out don't understand why b has to be 0 or even what is the point of bringing this up.

and right below that is:
{p ∈ P(F) : p(3) = 0}
is a subspace of P(F).

P(F) refers to the polynomial space. F is the set of fields and it contains C (complex numbers) and R (real numbers).

Again, what is the point of bringing this up and how do we know that p is a subspace of P(F) based off of the information given?

A subspace is itself a vector space again. Therefore it needs to contain $0$. But $0=(0,0,0,0) \in \{\,(x_1,x_2,x_3,x_4)\in \mathbb{F}^4\,;\,x_3=5x_4+b\,\}$ if and only if $b=0\,.$

I haven't looked it up, but I'm sure $\mathbb{F}$ stands for some field, not all fields, so $\mathbb{F}\in \{\,\mathbb{R},\mathbb{C}\,\}$ or (later on) any other field.

 

[Stephen Tashi]

Again, what is the point of bringing this up and how do we know that p is a subspace of P(F) based off of the information given?

Apparently "p(3) =0" indicates that x = 3 is a root of the polynomial. The sum of two such polynomials is a polynomial that satisfies that property and a scalar multiple of such a polynomial is a polynomial that satisfies that property.

From a didactic point of view, vector spaces defined as a set of polynomials are an important example because they provide an illustration that vector doesn't have to denote a physical quantity "with magnitude and direction". Presumably, students are already familiar with polynomials.

The set of polynomials (in one "indeterminate" x, with the usual definitions of "+" as an operation on polynomials and multiplication by a number as scalar multiplication ) is a good example of an infinite dimensional vector space.

The set of polynomials that have the root x = 3 is a good example of an infinite dimensional subspace.

The set of polynomials of degree at most 3 is a good example of a finite dimensional subspace.

The fact we can multiply two polynomials (by the usual definition of how to do so) illustrates that it may be possible to define operations other than "+" on two vectors. On the other hand, the axioms for a vector space do not define such operations. So some examples of vector spaces are "more than a vector space".

 

[mathwonk]

subspaces are flat sets that pass through the origin. being flat means all exponents in the equation have degree ≤ 1 (and at least one exponent is = 1), and passing through the origin means all exponents are equal to 1, i.e. no non zero constant term.