# Subspaces in Axler's Famous Textbook

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• gibberingmouther
In summary, the conversation discusses the conditions for a set to be a subspace, specifically in the cases of {(x1,x2,x3,x4) ∈ F4 : x3 = 5x4 + b} and {p ∈ P(F) : p(3) = 0}. The concept of subspace is important in understanding vector spaces, and examples such as polynomials demonstrate this. Subspaces are defined as flat sets that pass through the origin, and the conditions for a set to be a subspace are that it is closed under addition and scalar multiplication, and that it contains the zero vector.

#### 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?

gibberingmouther said:
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.

• gibberingmouther
gibberingmouther said:
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".

• gibberingmouther
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.

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• gibberingmouther