Matrix Transformations
Bayani Gaba-gaba
Every 2 by 2 matrix secretly describes a way of moving every point in the plane at once: multiply a point's coordinates by the matrix and out comes its image after a reflection, a rotation, an enlargement, or a shear. Recognising which matrix produces which transformation, and combining several transformations into a single matrix, turns geometry into pure algebra.
This topic builds directly on matrix multiplication: you will meet the standard matrices for reflections in the coordinate axes and the lines y = x and y = -x, rotations about the origin, enlargements centred at the origin, and shears, then combine transformations by multiplying their matrices together.
Manufura
- Understand transformations of the plane associated with 2 x 2 matrices, including reflections, rotations about the origin and enlargements with centre at the origin
- Find and use combination of transformations using matrix multiplication
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Nazarin Darasi
Barka da kammala darasi akan Matrix Transformations. Yanzu da kuka bincika mahimman ra'ayoyi da ra'ayoyi, lokaci yayi da zaku gwada ilimin ku. Wannan sashe yana ba da ayyuka iri-iri Tambayoyin da aka tsara don ƙarfafa fahimtar ku da kuma taimaka muku auna fahimtar ku game da kayan.
Za ka gamu da haɗe-haɗen nau'ikan tambayoyi, ciki har da tambayoyin zaɓi da yawa, tambayoyin gajeren amsa, da tambayoyin rubutu. Kowace tambaya an ƙirƙira ta da kyau don auna fannoni daban-daban na iliminka da ƙwarewar tunani mai zurfi.
Yi wannan ɓangaren na kimantawa a matsayin wata dama don ƙarfafa fahimtarka kan batun kuma don gano duk wani yanki da kake buƙatar ƙarin karatu. Kada ka yanke ƙauna da duk wani ƙalubale da ka fuskanta; maimakon haka, ka kallesu a matsayin damar haɓaka da ingantawa.
- Which matrix represents a reflection in the y-axis? A) [[1,0],[0,-1]] B) [[-1,0],[0,1]] C) [[0,1],[1,0]] D) [[-1,0],[0,-1]] Answer: B
- Which matrix represents a 90 degree anticlockwise rotation about the origin? A) [[0,1],[-1,0]] B) [[0,-1],[1,0]] C) [[-1,0],[0,-1]] D) [[1,0],[0,1]] Answer: B
- The matrix [[4,0],[0,4]] represents: A) a rotation of 90 degrees B) a reflection in the x-axis C) an enlargement, scale factor 4, centre the origin D) a shear Answer: C
- If R is carried out first, followed by M, the matrix for the combined transformation is: A) RM B) MR C) R + M D) M - R Answer: B
- A matrix that represents a shear parallel to the x-axis has the form: A) [[1,0],[k,1]] B) [[1,k],[0,1]] C) [[k,0],[0,k]] D) [[0,1],[1,0]] Answer: B
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