Vector Spaces

A vector is a cluster of numbers $(3, 2, 4)$. A vector space $V$ is a set of vectors for which the following rules are true:

• You can multiply one of the vectors by a real number and it will result in another vector in the vector space. The space is closed by multiplication.
• Multiplying by $1$ does nothing to a vector. $1\vec{v} = \vec{v}$
• You may add two of the vectors in the space and it will result in another vector in the space. The space is closed by addition.
• Adding vectors does not depend on the order in which you do it. Adding many vectors finds a sum of the vectors which is always results in the same vector.
• There exists a vector in the space, referred to as the zero vector, which does nothing when added to other vectors. $\vec{v} + 0 = \vec{v}$ where $0 \in V$
• All vectors have a negative version, and the sum of a vector and its negative is the zero vector. $-\vec{v} + \vec{v} = 0$
• Three more...

Two vectors are linearly dependent if one is a multiple of the other, meaning that they form parallel vectors. $(1,2,3)$ and $(-2, -4, -6)$ are linearly dependent vectors. Another way to define linearly dependent vectors is that you can achieve zero with scalar multiples and vector addition. $2(1,2,3) + (-2,-4,-6) = 0$. Three vectors are linearly dependent if one of the vector exists on the plane created by the other two vectors. So three vectors are also linearly dependent if you can achieve zero with scalar multiples and addition.

One better mechanism to check if vectors are linearly dependent is by finding a determinant. Take three vectors $v_1$, $v_2$, and $v_3$, and create a matrix with each of the vectors as a column.

$ \begin{matrix} 1 & 2 & 3\ a & b & c \end{matrix} $

https://math.okstate.edu/people/binegar/3013-S99/3013-l12.pdf YouTube series Essence of Linear Algebra