Chapter-12-2: Column space A image

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2022-12-08 08:12

This example demonstrates the concept of column space in linear algebra, visualizing vector spaces through geometric figures. Key mathematical concepts include linear transformations, vector spaces, and matrix operations. Construction involves GeoGebra commands such as Intersect, PointIn, Vector, etc. Suitable for high school students or college students learning linear algebra basics, helping to understand abstract concepts.

GeoGebra Commands & Parameters

Intersect(xAxis, zAxis) → A
PointIn(xOyPlane) → B
Vector(A, B) → u
Vector(A, C) → v
Vector(A, D) → w
Vector(C, D) → a
Angle(D, C, A) → α
Vector(B, D) → b
Segment(B, D) → f
Segment(C, D) → g
LaTeX(err1, true, true) → text1
Segment((0, 0, 0), C) → h
Surface(u_{1}, v_{1}, 0, u_{1}, -5, 5, v_{1}, -5, 5) → c
Text("$Col(\textit{A})$", E, false, true) → text3
Point(h) → G
Text("$\textit{A}\hat{\mathbf x}=\text{Proj}_{Col(\textit{A})}(\mathbf b)$", G + (-0.5, -1.8, 0.35), false, true) → text4

AI Drawing Prompt Reference

Draw a 3D space image of the column space, including the x-axis, y-axis, z-axis, and origin O. Use vectors u, v, and w to represent the column space of matrix A, vector a to represent the vector from C to D, and angle α to represent the angle between vectors C and D. Draw vector b and line segments f and g, as well as the line segment from B to D. In the 3D space, draw a plane c generated by vectors u and v. Label point G to indicate that the projection of matrix A on the x-axis equals the projection of vector b on the column space of A.

The prompt above was compiled from the original drawing commands and is for reference only.