Vector Spaces

• A vector spaces is a set of vectors

• Addition and Scalar multiplication can be used!

• Vector spaces can be anything, vectors, polynomials, functions

• If I have two polynomials of the first degree , u = x + 1 and v = 2x - 3, they both share the same vector space, because they are same degree

• If I add the polynomials u = x + 1 and v = 2x - 3 together, I get uv = 3x - 2

• The resulting uv = 3x - 2 is also a polynomial of degree one, so we’re still in that original vector space!

• The same is true if I multiplied u = x + 1 by 5. I would get 5u = 5x +5. 5u is still a polynomial of degree one, so we’re still in that first degree polynomial space!

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• The same is true of I have the two matrices

$$\hat{u}=\left[\begin{array}{cc}2& 1\\ 1& 3\end{array}\right]$$

and

$$\hat{v}=\left[\begin{array}{cc}-1& 0\\ 1& 1\end{array}\right]$$

• If I add and v̂, I get û + v̂

$$\hat{u}+\hat{v}=\left[\begin{array}{cc}1& 1\\ 2& 4\end{array}\right]$$

• This results in a 2D vector, meaning that we are staying in the same 2D vector space.
• 5û would result in the same thing, another 2D vector. We’re good here.

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These are known as the properties of closure.

• Given ûεV (û is a vector in the vector space V) and scalar C, then Cûε V (The multiplied scalar will also be in vector space V)

• Given ûεV and v̂εV , then û + v̂ϵ V (The new added vector is also in vector space V!)

The dimension of the vector is usually noted by ℝⁿ. The amount of non zero rows in a vector will give you that.

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What are Axioms? What are the 8 Axioms?

Axioms are a bunch of rules which hold true inside of a vector space. There are 10 of which hold true in any vector space V

With vectors û, v̂ and ŵ, and scalars a and b

Addition

1. û + v̂ = v̂ + û This is known as the commutative property of addition!

2. û + (v̂ + ŵ) = (û + v̂) + ŵ This is known as the associative property

3. û + 0 = û + 0 There is a zero vector. This is known as the additive identity

4. û + û' For every vector in a vector space, there is another vector that can bring it back to zero. This property is known as the additive inverse! Multiplication

5. For ûϵ**V* 1û = u For every single vector in the vector space V, 1* û will equal û. This is known as the multiplicative identity

6. a * (bu) = (ab) * u This property does not seem to have a name, but is used to combine scalars. Scalars can be combined as such.

7. (a + b)u = au + bu This is known as distributivity of scalar multiplication with respect to field addition. Vectors can distribute like this!

8. a(u+v) = au + av This is known as distributivity of scalar multiplication with respect to vector addition. Just like the last property, this works as well! 

Vector Subspaces

• Vector Subspaces are, you guessed it, a vector space inside of another vector space.

• If there is a subspace S inside of a vector space V, the subspace S must follow all of V’s rules.

• The earlier Addition and Scalar multiplication rules ([properties of closure](Final Linear Algebra Review)) can be used to ensure conformance to the new subspace

Plane in a Three-Dimensional Space

• If you know that you have a plane in a 3D space, you DON’T know that it immediately is a subspace.
• This is because of the possibility of having a scalar of “0” creating a “zero vector”.
• If you know that the plane passes through (0,0,0), you’re good, because that condition works.
• If you don’t know that it passes through the origin (0,0,0), you don’t have enough information to determine whether the plane is a subspace of the 3D space.

Null Space

• Null Space is a situation to where all of the vectors in AX=B, are AX=0
• Null Space is noted by N(A) (A being the matrix)
• This occurs when you have a A matrix, transformed by an “X vector” to make it equal to zero.

$$A=\left[\begin{array}{cc}2& 1\\ 1& 3\end{array}\right]$$ $$\left[\begin{array}{cc}2& 1\\ 1& 3\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=B$$

To solve for the X’s which make B=0, we can set up a homogenous solution with matrix augmentation.

$$\left[\begin{array}{cc}2& 1|0\\ 1& 3|0\end{array}\right]=B$$

• Do REF, or RREF, and solve for the unknown X’s! (keep in mind, this matrix is just:

$$2x_1+x_2=0$$ $$1x_1+3x_2=0$$

)

A weird thing to think about, is that the null space is actually a subspace of A! It meets all of the criteria!

• What is happening, is that the subspace is being crushed into zero by a transformation, and becomes a zero vector