4.2: Pre-class assignment review
- Page ID
- 63856
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Vector Addition Properties
For any vectors \(x\), \(y\), and \(z\) of the same size/dimension, we have the following properties:
- Vector addition is commutative: \(x + y = y + x\).
- Vector addition is associative: \((x + y) + z = x + (y + z)\). We can therefore write both as \(x + y + z\).
- Adding the zero vector to a vector has no effect: \(x + 0 = 0 + x = x\). (This is an example where the size of the zero vector follows from the context, i.e., its size must be the same as the size of \(x\))
- \(x − x = 0\). Subtracting a vector from itself yields the zero vector. (Here too the size of 0 is the size of a.)
Scalar-Vector Multiply Properties
For any vectors \(x\), \(y\), and scalars \(a\), \(b\), we have the following properties
- Scalar-vector multiply is commutative: \( ax = x * a \); This means that scalar-vector multiplication can be written in either order.
- Scalar-vector multiply is associative: \( (ab)x = a(bx) \)
- Scalar-vector multiply is distributive: \( a(x + y) = ax + ay \), \( (x+y)a = xa + ya \), and \( (a+b)x = ax + bx \).