Subspace projection matrix example | Linear Algebra | Khan Academy

Example of a transformation matrix for a projection onto a subspace

Missed the previous lesson?

Linear Algebra on Khan Academy: Have you ever wondered what the difference is between speed and velocity? Ever try to visualize in four dimensions or six or seven? Linear algebra describes things in two dimensions, but many of the concepts can be extended into three, four or more. Linear algebra implies two dimensional reasoning, however, the concepts covered in linear algebra provide the basis for multi-dimensional representations of mathematical reasoning. Matrices, vectors, vector spaces, transformations, eigenvectors/values all help us to visualize and understand multi dimensional concepts. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.

About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.

For free. For everyone. Forever. #YouCanLearnAnything

[UnwrittenLines]

Thanks this helped so much. I can't express my gratitude. Keep up the good work.

[rbmherbert]

this is super helpful- thank you so much!

[s.gorelski]

The basis set for A is not orthogonal here. The dot product of [1 0 0 1] and [0 1 0 1] is one, not zero. So, all those remaining calculations are faulty.

[eliordadon]

if X has d+1 columns, why the sum of all the elements in P are equal to d+1?

why 0<=P_m, m <=1, m=1, ..., N
why -0.5 <= P_n, m <= 0.5, n!=m
?

[ryanjin]

What if you try to project R5 onto this subspace? How would you multiply the 1x5 vector by a 4x4 matrix

[saca]

what software are u using to write with?

[aamailhot]

this is the foundation of science here...which means boring but useful. deal with it, freshman.

[ABIYSCGSTEGD]

This video is #112 in this series. Why is time wasted working out every step of the dot product and matrix multiplication? After 111 videos, don't students know this? This makes this video way longer than it needs to be and takes the focus off the main ideas.