Common Vector Spaces

Common Vector Spaces

We will now look at proofs of some commonly recognized vector spaces. We will be utilizing the ten axioms that define a vector space thoroughly in these proofs, so feel free to consult the axioms on the Vector Spaces page whenever necessary!

The Zero Vector Space

The zero vector space represents a set whose only element is $0$, to which we define addition and scalar multiplication in the standard way. We will go forward to prove that this set is in fact a vector space under these operations:

  • Proof: Since $0$ is the only element in this vector space, axioms 1-4 and 6-9 should be self-evident. Axioms 5 and 10 are rather self evident, but we will verify them too.
  • Axiom 5 is verified since $0$ is the only element in the zero vector space, so clearly the zero vector space is closed under addition.
  • Axiom 10 is verified since once again, $0$ is the only element in the zero vector space. Any scalar $k$ multiplied by $0$ will be $0$, so the zero vector space is closed under multiplication.
  • Since the zero vector space is a nonempty set containing defined operations of addition and multiplication, and all ten axioms hold, then the zero vector space is proven to in fact be a vector space. $\blacksquare$

Vector Spaces of m x n Matrices

The vector space denoted $M_{mn}$ represents the set of $m \times n$ matrices containing real number entries where standard matrix addition and standard scalar matrix multiplication are defined. Let's now prove that $M_{mn}$ is in fact a vector space given these operations:

  • Proof: To prove $M_{mn}$ is a vector space, we first note that axioms 1-4 and 6-9 already hold when addition is defined to be standard matrix addition and multiplication is defined to be standard scalar matrix multiplication. We now need to show that axioms 5 and 10 hold.
  • Axiom 5 says that for $M_{mn}$ to be a vector space, the objects in $M_{mn}$ must be closed under addition. Suppose that $A$ and $B$ are both $m \times n$ matrices. The sum $A + B$ is also an $m \times n$ matrix so clearly $M_{mn}$ is closed under addition.
  • Axiom 10 says that for $M_{mn}$ to be a vector space, the objects in $M_{mn}$ must be closed under multiplication. Suppose that $k$ is a scalar and $A$ is an $m \times n$ matrix. The product $kA$ is still an $m \times n$ matrix as we take all entries in $A$ and multiply them by the scalar $k$ to get $kA$. Therefore, $M_{mn}$ is closed under multiplication.
  • Since $M_{mn}$ is a nonempty set containing defined operations of addition and multiplication, and all ten axioms hold, then $M_{mn}$ is a vector space. $\blacksquare$

Vector Spaces of Real-Valued Functions

The vector space denoted $F(-\infty, \infty)$ represents the set of real-valued functions defined on the interval $(-\infty, \infty)$ where for two functions $f$ and $g$, addition is defined to be the sum $f + g$, and for some scalar $k$, multiplication is defined to be $kf$. We will now verify that $F(-\infty, \infty)$ defined with these operations is a vector space:

  • Proof: We will go through all ten axioms to ensure that $F(-\infty, \infty)$ is indeed a vector space.
  • Axioms 1 and 2 are self evident for real-valued functions, that is $f + g = g + f$ and $(f + g) + h = f + (g + h)$.
  • Axiom 3 can be verified with the zero constant function that we'll denote as $0(x) = 0$. We note that $f + 0 = 0 + f = f$ for all $f \in F(-\infty, \infty)$.
  • Axiom 4 is verified for a function $f$ by reflecting it about the $x$-axis to obtain the real-valued function $-f$, that is $f + (-f) = 0$.
  • Axiom 5 is verified because the sum of two real-valued functions $f + g$ is still a real-valued function, so $F(-\infty, \infty)$ is closed under addition.
  • Axiom 6-8 are self evident as scalar multiples of a function stretch/compress said function.
  • Axiom 9 is verified by the scalar $1$, that is $1f = f\cdot 1 = f$.
  • Axiom 10 is verified because a scalar multiple $k$ of a real-valued function $f$ is still a real-valued function, so $F(-\infty, \infty)$ is closed under multiplication.
  • Since $F(-\infty, \infty)$ is a nonempty set containing defined operations of addition and multiplication and all ten axioms hold, then $F(-\infty, \infty)$ is a vector space. $\blacksquare$
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