Term
| When do can the lines of a 2D graph imply about the system that produced them? |
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Definition
They tell us something about the solution to the system as
- lines that intersect indicate there is one solution
- parallel lines indicate there is no solution
- a single line indicates there are many solutions
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Term
| What is a vector in geometric terms? |
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Definition
A vector is an object haveing a magnitude and direction. i.e. it has a length and points in some direction. |
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Term
| When are vectors considered equal? |
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Definition
When the have the same length and point in the same direction. |
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Term
| When is a vector incanonical position? |
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Definition
When it is drawn at the origin (0,0) to some point (x,y). |
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Term
| What is the canonical name of a vector? |
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Definition
The canonical name is the endpoint of a vector when it is drawn in canonical position.
For example, [image] is the canonical name of a vector drawn from (0,0) to (1,2) |
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Term
| What is a direction vector? |
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Definition
A vector that describes the displacement of a line through one or more points.
For example, a line running through the points (1,2) and (3,1) has a direction vector of (2,-1)
[image] |
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Term
Translate a plane described by a singular linear system
[image]
to a vector description |
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Definition
Think of it as a one equation linear system and parameterize
2x + y + z = 4 -> x = 2- y/2 - z/2
so that
[image] |
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Term
| General description of a k-dimensional linear surface(k-flat) |
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Definition
[image]
where [image] and [image]
A line represents a one dimensional surface; a plane, a two dimensional surface, etc. |
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Term
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Definition
A linear surface with one less dimension than the space it occupies i.e. when in [image] we have a (n - 1) flat.
For example, a hyperplane H in [image] is a line; a hyperplane H in [image] is a plane |
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Term
| What is the solution set of a linear system with n-unknowns? |
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Definition
The solution set is a linear surface in [image] meaning it is a k-dimensional linear surface where k is the number of free variables in an echelon form version of the system. |
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Term
| What is the solution set of a homogeneous system? |
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Definition
A linear surface passing through the origin. |
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Term
| In a general solution, what is the effect of the particular solution? |
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Definition
The particular solution represents how the homogeneous solution is offset from the origin. |
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Term
| Angle between a diagonal and the x-axis |
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Definition
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Term
| Midpoint of a line segment |
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Definition
[image]
The above can be generalized for any n.
[image] |
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Term
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Definition
The angle between two non-zero vectors [image] is the arc cosine of the dot product of the two vectors divided by the product of their lengths.
[image]
if either vector is a zero-vector 'we take the angle to be right' (90°) i.e. vectors are perpendicular (orthogonal) when their dot product = 0.
Vectors are parallel if their dot product = the product of their lengths. |
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