Linear combinations, span, and basis vectors

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Consider each vector coordinate as scalar, if you have a vector with coordinate $mat(2,1)$, the resultant vector is the sum of 3 X the unit vector in the X axis, $\hat{i}$ and -2 X the unit vector in the Y axis, $\hat{j}$.

Adding two scalar vectors of different dimension, provides a new vector in both dimension.

By scaling and adding 2 vectors of different dimension you can have every possible 2 dimensional vector. You can reach any point in the 2 dimensional plane.

• The $\hat{i}$ and $\hat{j}$ are the basis vector, or the unit vectors.
• Any time we add 2 vectors like a$\vec{v}$+b$\vec{w}$, is called Linear Algebra.
• span - list of all possible vector, that you can reach with a linear combination of given pair of vectors
• if you fix one unit vector and move the other vector, the tip of the result vectors would be aligned into a straight line,
• if you move both the vectors,
  • if both the vectors are 0, the span is a point, cause whatever you multiply with 0 it is still 0, the resulting vector will have no magnitude
  • if both vectors directions are aligned, then span is a straight line
  • if both vectors have different direction, then the span a plane
• The span of 2 vectors in a 3 dimensional space is a plane.
• The span of 3 vectors in a 3 dimensional space is a
  • if the 3 rd vector is on the span of the first 2, the span will remain the same.
  • if not, you get access to the entire 3 dimensional array
• one way to think is, the 3rd vector, can move the span of the 2 vectors through its space.

Linearly Dependent

• when a vector doesn’t change the span of a vector, those 2 vectors are said to be linearly dependent
• A linearly dependent vector can be expressed as a linear combination of the other vectors, since it is already part of their span
• Linearly Independent - other vectors are
• Basis of a vector space - set of linearly independent vectors that span the entire vector space.