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 otherTry 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.