Spatial orthogonal and vector projection

4 Sub-spaces:

1. Row space, 0 space
2. Column space, a transpose of 0 space.

What is an orthogonal vector? The two vectors have an angle of 90 degrees.

Right angle means: X + y Gets the new vector = x + y

|x|^2 + |y|^2 = |x+y|^2

How to prove it?

X ' *x + y ' *y = (x+y) ' * (x+y)

= (x ' +y ') * (x+y)

=x*x ' + y*y ' +x ' *y+x*y '

launched X ' *y=0

So x is perpendicular to Y.

SubSpace, sub-space. SubSpace quadrature means that any vector in S is orthogonal to any vector in T.

The vertical walls and floors are not orthogonal, as they have a cross line. Two spaces if orthogonal, only intersect at 0 points. So the line space and the 0 space are orthogonal. Because Ax=0, where a is a row space, X is a vector of 0 space.

And the number of their bases is exactly equal to N, so they are complementary.

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Let's look at two orthogonal examples of two vectors:

Where A and B are two vectors, X is a number, and x*a is a projection of B on a.

Can get a ' * (b–x * a) = 0

And then get a ' *b–a ' *x*a=0 because x is constant, so

A ' * b = a ' *a*x

Both sides multiply 1/a *1/a ', noticing that the order of a and a ' is reversed.

Get x = A ' *b/(A*a ').

Spatial orthogonal and vector projection