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## Linear algebra

### Unit 2: Lesson 2

Linear transformation examples

# Introduction to projections

Determining the projection of a vector on s line. Created by Sal Khan.

## Want to join the conversation?

• I don't see how you're generalizing from lines that pass thru the origin to the set of all lines. Wouldn't it be more elegant to start with a general-purpose representation for any line L, then go fwd from there? Seems like this special case is missing information....positional info in particular.
• You can get any other line in R2 (or RN) by adding a constant vector to shift the line. Transformations that include a constant shift applied to a linear operator are called affine. Note, affine transformations don't satisfy the linearity property.
• Explain projection of a vector
(1 vote)

The shadow is the projection of your arm (one vector) relative to the rays of the sun (a second vector).
• Just a quick question, at you cannot cancel the top vector v and the bottom vector v right? Is this because they are dot products and not multiplication signs?
• You get a different answer (a vector divided by a vector, not a scalar), and the answer you get isn't defined.
• why are you saying a projection has to be orthogonal?
• to use Sal's method, then "x - cv" must be orthogonal to v (or cv) to get the projection. Why? because if x and v are at angle t, then to get ||x||cost you need a right triangle
(1 vote)
• Please remind me why we CAN'T reduce the term (x*v / v*v) to (x / v), like we could if these were just scalars in numerator and denominator... but we CAN distribute ((x - c*v) * v) to get (x*v - c*v*v) ?
Where do I find these "properties" (is that the correct word? i.e. what I can and can't transform in a formula), preferably all conveniently** listed?
Presumably, coming to each area of maths (vectors, trig functions) and not being a mathematician, I should acquaint myself with some "rules of engagement" board (because if math is like programming, as Stephen Wolfram said, then to me it's like each area of maths has its own "overloaded" -, +, * operators. But where is the doc file where I can look up the "definitions"??).
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**i.e. without diving into Ancient Greek or Renaissance history ;)_
• For example, does:

(u dot v)/(v dot v) = ((1, 2)dot(2, 3))/((2, 3)dot(2, 3)) = (1, 2)/(2, 3)?

No. The quotient of the vectors u and v is undefined, but (u dot v)/(v dot v) is.
(1 vote)
• At , how can you multiply vectors such a way? They are (2x1) and (2x1). Can they multiplied to each other in a first place?
• What does orthogonal mean?
• In an inner product space, two elements are said to be orthogonal if and only if their inner product is zero. In Euclidean n-space, Rⁿ, this means that if x and y are two n-dimensional vectors, then x and y are orthogonal if and only if x · y = 0, where · denotes the dot product.
• hi there, how does unit vector differ from complex unit vector?
• Many vector spaces have a norm which we can use to tell how large vectors are. R^2 has a norm found by ||(a,b)||=a^2+b^2. The complex vectors space C also has a norm given by ||a+bi||=a^2+b^2. The look similar and they are similar. Unit vectors are those vectors that have a norm of 1. There is a pretty natural transformation from C to R^2 and vice versa so you might think of them as the same vector space. But they are technically different and if you get more advanced with what you are doing with them (like defining a multiplication operation between vectors) that you want to keep them distinguished.
• Why not mention the unit vector in this explanation?
• v actually is not the unit vector. The unit vector for L would be (2/sqrt(5), 1/sqrt(5)) .

If you want to solve for this using unit vectors here's an alternative method that relates the problem to the dot product of x and v in a slightly different way:

First, the magnitude of the projection will just be ||x||cos(theta), the dot product gives us x dot v = ||x||*||v||*cos(theta) , therefore ||x||*cos(theta) = (x dot v) / ||v|| . This gives us the magnitude so if we now just multiply it by the unit vector of L this gives our projection (x dot v) / ||v|| * (2/sqrt(5), 1/sqrt(5)) . Which is equivalent to Sal's answer.