Vector Fundamentals Quiz

Understanding Vectors: From Arrows in Space to Lists of Numbers

• There are three perspectives on vectors: the physics student perspective, the computer science student perspective, and the mathematician's perspective.
• In physics, vectors are arrows pointing in space, while in computer science, they are ordered lists of numbers.
• Mathematicians generalize both views, saying that a vector can be anything where there's a sensible notion of adding two vectors and multiplying a vector by a number.
• In linear algebra, vectors always have their tail at the origin of a coordinate system.
• In two dimensions, a vector's coordinates are a pair of numbers indicating how far to walk along the x-axis and parallel to the y-axis.
• In three dimensions, a vector's coordinates are a triplet of numbers indicating how far to move along the x-axis, y-axis, and z-axis.
• Vector addition involves moving the tail of the second vector to the tip of the first vector and drawing a new vector from the tail of the first vector to the tip of the second vector.
• Scalar multiplication involves scaling a vector by a number, which can stretch, squish, or reverse the direction of the vector.
• Linear algebra revolves around vector addition and scalar multiplication, which are fundamental operations that mathematicians think about independently and abstractly.
• Linear algebra allows for translating between the geometric and numerical representations of vectors.
• Linear algebra is useful for conceptualizing many lists of numbers in a visual way and describing space and the manipulation of space using numbers that can be crunched and run through a computer.
• The next video will cover concepts such as span, bases, and linear dependence.