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- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/03%3A_Linear_Transformations_and_Matrix_Algebra/3.00%3A_Prelude_to_Linear_Transformations_and_Matrix_AlgebraThis page explores the non-linear complexities of a robot arm's joint movements and hand positions through the transformation function f(θ,ϕ,ψ). It discusses the relationship between ma...This page explores the non-linear complexities of a robot arm's joint movements and hand positions through the transformation function f(θ,ϕ,ψ). It discusses the relationship between matrices and transformations, covering how transformations can be expressed with matrices, their properties, and how matrix multiplication relates to composition. The chapter culminates in understanding matrix arithmetic and solving matrix equations.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/02%3A_Systems_of_Linear_Equations-_Geometry/2.8%3A_Bases_as_Coordinate_SystemsThis page explains how a basis in a subspace serves as a coordinate system, detailing methods for computing B-coordinates and converting to standard coordinates. It illustrates finding a...This page explains how a basis in a subspace serves as a coordinate system, detailing methods for computing B-coordinates and converting to standard coordinates. It illustrates finding a basis through row reduction, using examples to demonstrate the representation of vectors as linear combinations of basis vectors. Visual aids support the explanations, emphasizing the verification of linear independence and span to confirm a basis.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/09%3A_Change_of_Basis/9.01%3A_The_Matrix_of_a_Linear_TransformationThe fact that T=C−1DTACB means that the action of T on a vector v in V can be performed by first taking coordinates (that is, applying CB to v), th...The fact that T=C−1DTACB means that the action of T on a vector v in V can be performed by first taking coordinates (that is, applying CB to v), then multiplying by A (applying TA), and finally converting the resulting m-tuple back to a vector in W (applying C−1D).
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.11%3A_The_Matrix_of_a_Linear_Transformation/7.11E%3A_Exercises_for_Section_7.11This page presents exercises on linear transformations and matrix representations across vector spaces like R2, P2, and M22. It includes finding coordinate vectors, ...This page presents exercises on linear transformations and matrix representations across vector spaces like R2, P2, and M22. It includes finding coordinate vectors, matrices under specified bases, and exploring the kernel and image of transformations. Exercises feature transformations, such as differentiating polynomials and mapping 2×2 matrices to R2, complete with specified bases and matrix representations.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/01%3A_Systems_of_Equations/1.01%3A_Systems_of_Linear_Equations/1.1E%3A_Exercises_for_Section_1.1This page offers exercises on solving linear systems graphically, focusing on finding intersection points of lines, understanding different solution scenarios (no, unique, infinite solutions), and exa...This page offers exercises on solving linear systems graphically, focusing on finding intersection points of lines, understanding different solution scenarios (no, unique, infinite solutions), and examining common intersections of multiple lines or planes. It includes a word problem involving weights of four individuals and tasks requiring the construction of linear systems with defined solution properties and graphical relationships.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_I%3A_Differential_Calculus/05%3A_Differential_Calculus_with_Parametric_Curves/5.03%3A_Chapter_5_Review_ExercisesSketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. x=1+t,y=t2−1,−1≤t≤1 x=et,y=1−e3t,0≤t≤1 For, \(x=\ln(t),\; y=t^2−...Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. x=1+t,y=t2−1,−1≤t≤1 x=et,y=1−e3t,0≤t≤1 For, x=ln(t),y=t2−1,t=1, find the equation of the tangent line to the given curve. Find dydx,dxdy, and d2xdy2 of y=(2+e−t),x=1−sint dydx=1etcost,dxdy=etcost,d2xdy2=e2t(sint−cost).
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.08%3A_The_Matrix_of_a_Linear_Transformation_IIIn this section we learn how to represent a linear transformation with respect to different bases.
