HackerWhat does a matrix do, and which outputs can disappear?
Read a column as an arrow, then a matrix as an action on that arrow. Solving an equation becomes a question about reachable targets, giving collapse, eigenvectors and rotations a geometric interpretation.
Questions in this reading · 8
- What do the numbers in a vector describe?
- What does multiplying a vector mean?
- What is the same in a diagram and a column calculation?
- How can columns explain a matrix equation?
- Which equations survive a collapse in dimension?
- Which outputs disappear when a transformation collapses space?
- Why shift a matrix before looking for an eigenvector?
- How do rotations change in four dimensions?
What do the numbers in a vector describe?
Translate between an arrow and a list of coordinates. Each representation makes a different aspect of the same object easier to inspect.
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the geometric understanding is what lets you judge what tools to usea language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.What does multiplying a vector mean?
Scaling one vector and combining two vectors are different operations. Check the operands and the type of result before interpreting the notation.
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these two fundamental operations, vector addition and scalar multiplication.multiplying two vectors together is the dot product plus the wedge productWhat is the same in a diagram and a column calculation?
Switch between geometric motion and coordinate arithmetic to see one linear operation from two angles. Strang’s column combination is a worked example; neither representation is best for every task.
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the ability to translate back and forth between themthe geometric understanding is what lets you judge what tools to useA times x is a combination
of the columns of A.How can columns explain a matrix equation?
Translate coordinates into a geometric picture. The columns of a matrix then show which combinations of vectors can reach the desired output.
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the ability to translate back and forth between themlinear combinations of the columns fill three dimensional spaceWhich equations survive a collapse in dimension?
Once a transformation flattens space, reachable outputs lie in its image. A target inside that image and one outside it lead to different answers.
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A determinant of 0 would mean that all of space is squished onto something with 0 volumeIf these three columns
all lie in the same plane, then their combinations
will lie in that same plane.Which outputs disappear when a transformation collapses space?
Some target vectors fall outside the image after a collapse. In the eigenvalue example it is A minus lambda times the identity that becomes singular; this does not make every system unsolvable.
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A determinant of 0 would mean that all of space is squished onto something with 0 volumeIf these three columns
all lie in the same plane, then their combinations
will lie in that same plane.make this determinant zero, meaning the tweaked transformation squishes space into a lower dimensionWhy shift a matrix before looking for an eigenvector?
A zero determinant signals collapse into a lower dimension. Applying that test to the shifted matrix explains how a nonzero vector can lie in its null space.
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A determinant of 0 would mean that all of space is squished onto something with 0 volumemake this determinant zero, meaning the tweaked transformation squishes space into a lower dimensionHow do rotations change in four dimensions?
Compare a fixed axis in three dimensions with planes of rotation in four. The familiar axis description does not carry over unchanged.
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what you have found is the axis of rotationtwo separate 2D rotations, happening perpendicular to and in sync with each otherWhy does an exponential trace a circle?
Return to the plane and separate position from the rule for moving. If velocity equals position multiplied by i, it is perpendicular to the radius and has the radius’s length. Starting at one gives motion around the unit circle at unit angular speed. This account explains how a local rule determines a trajectory.
Passages and timestamps · 1
the velocity at that time will be a 90 degree rotation of that position.Now ask whether moving through angle a and then b is the same as moving through a+b. The group-theoretic account starts with composition: adding inputs corresponds to multiplying outputs. One account gives the velocity; the other lets you check how two actions combine. A single angle works beautifully here. It does not follow that one angle specifies every rotation in four dimensions.
Passages and timestamps · 1
adding the inputs corresponds with multiplying the outputsTry it on your own example
Choose a matrix that flattens the plane. Draw one reachable and one unreachable target, then check both by combining its columns.