[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-detail-572985-en":53,"doc-seo-572985-105":74},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":5,"data":54},{"doc_id":55,"user_id":56,"nickname":57,"user_avatar":58,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":60,"doc_content":61,"file_id":62,"file_url":63,"file_type":64,"file_size":65,"view_count":66,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":9,"language":67,"language_code":68,"site_id":69,"html_lang":68,"table_of_contents":70,"faqs":71,"seo_title":72,"seo_description":60,"update_tm":73,"read_time":4},572985,962090768927,"Muffet","https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","Common Matrix Transformations - Mathematical Guide","This document provides a comprehensive reference guide to common matrix transformations in linear algebra. It details various geometric operations, including identity transformations, axial reflections, rotations by specific angles, and scaling enlargements. Each transformation is accompanied by its corresponding matrix representation and mapping notation, facilitating a clear understanding of how these mathematical operations affect coordinates in a 2D plane. The content serves as a foundational resource for students and practitioners needing to visualize and apply matrix-based transformations for geometric manipulation and coordinate geometry analysis.","Common Matrix Transformations  \n\n| [􀫚􀫙 􀫙􀫚] | Identity matrix.\u003Cbr>Right remains right, up remains up. | [􀢞􀢟] ⟼ [􀢞􀢟] |\n| --- | --- | --- |\n| [ 1 01] | Reflection in the 􀝕-axis.\u003Cbr>Right has become left, up remains up. | [􀝔􀝕] ⟼ [ ] |\n| [10 1] | Reflection in the 􀝔-axis.\u003Cbr>Right remains right, up has become down. | [􀝔􀝕] ⟼ [] |\n| [1 1] | Rotation by 180°\u003Cbr>Right has become left, up has become down. | [􀝔􀝕] ⟼ [−􀝔􀝕] |\n| [01 10] | Reflection in the line 􀝕 = 􀝔 .\u003Cbr>Right has become up, up has become right. | [􀝔􀝕] ⟼ [􀝕􀝔] |\n| [01 1] | Rotation by 90° anticlockwise.\u003Cbr>Right has become up, up has become left. | [􀝔􀝕] ⟼ [] |\n| [ 1 10] | Rotation by 90° clockwise.\u003Cbr>Right has become down, up has become right. | [􀝔􀝕] ⟼ [ ] |\n| [1 1] | Reflection in the line 􀝕 = −􀝔 .\u003Cbr>Right has become down, up has become left. | [􀝔􀝕] ⟼ [−􀝕􀝔] |\n\n\n| [􀜽0 01] | Enlargement by scale factor 􀜽 in the 􀝔 direction. Right is multiplied by 􀜽, up remains up. | [􀝔􀝕] ⟼ [􀜽􀝔􀝕] |\n| --- | --- | --- |\n| [10 0􀜽] | Enlargement by scale factor 􀜽 in the 􀝕 direction. Right remains right, up is multiplied by 􀜽 . | [􀝔􀝕] ⟼ [􀝔􀜽􀝕] |\n| [􀜽0 0􀜽] | Enlargement by scale factor 􀜽 from the origin.\u003Cbr>Right is multiplied by 􀜽, up is multiplied by 􀜽 . | [􀝔􀝕] ⟼ [􀜽􀜽􀝔􀝕] |\n| [􀜽0 0􀜾] | Enlargement by scale factor 􀜽 in the 􀝔 direction and scale factor 􀜾 in the 􀝕 direction.\u003Cbr>Right is multiplied by 􀜽, up is multiplied by 􀜾 . | [􀝔􀝕] ⟼ [􀜽􀜾􀝔􀝕] |\n\nCombinations of these matrices give multiple transformations. For instance, two reflections generate a rotation.","cbCaiv0by0Up7vDb","https://ap.wps.com/l/cbCaiv0by0Up7vDb","pdf",290787,3,"English","en",105,"# Common Matrix Transformations\n## Basic Geometric Transformations\n## Enlargement and Scaling","[{\"question\":\"What is the purpose of an identity matrix in geometric transformations?\",\"answer\":\"The identity matrix leaves coordinates unchanged, ensuring that 'right remains right' and 'up remains up' in the transformation process.\"},{\"question\":\"How are rotations achieved using matrices?\",\"answer\":\"Rotations are performed by specific matrices, such as those for 90-degree clockwise, 90-degree anticlockwise, or 180-degree rotations, which map coordinates to new positions based on the rotation angle.\"},{\"question\":\"What is the effect of an enlargement matrix with scale factors a and b?\",\"answer\":\"An enlargement matrix with scale factors a and b scales the x-coordinate by a factor of a and the y-coordinate by a factor of b, effectively stretching or shrinking the object along the respective axes.\"}]","Common Matrix Transformations - Mathematical Guide | PDF",1791930382,{"code":4,"msg":75,"data":76},"ok",{"site_id":69,"language":68,"slug":77,"title":59,"keywords":78,"description":60,"schema_data":79,"social_meta":134,"head_meta":136,"extra_data":138,"updated_unix":139},"common-matrix-transformations-mathematical-guide","",{"@graph":80,"@context":133},[81,96,116],{"@type":82,"itemListElement":83},"BreadcrumbList",[84,88,91,93],{"item":85,"name":86,"@type":87,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":89,"name":10,"@type":87,"position":90},"https://docshare.wps.com/template/",2,{"item":92,"name":51,"@type":87,"position":66},"https://docshare.wps.com/template/general/",{"item":94,"name":59,"@type":87,"position":95},"https://docshare.wps.com/template/common-matrix-transformations-mathematical-guide/572985/",4,{"url":94,"name":59,"@type":97,"image":98,"author":103,"headline":59,"publisher":105,"fileFormat":108,"inLanguage":68,"description":60,"dateModified":109,"datePublished":110,"encodingFormat":108,"isAccessibleForFree":111,"interactionStatistic":112},"DigitalDocument",{"url":99,"@type":100,"width":101,"height":102},"https://docshare.wps.com/thumbnails/common-matrix-transformations-mathematical-guide/572985.png","ImageObject",442,249,{"name":57,"@type":104},"Person",{"url":85,"name":106,"@type":107},"DocShare","Organization","application/pdf","2026-10-17","2026-10-13",true,{"@type":113,"interactionType":114,"userInteractionCount":66},"InteractionCounter",{"@type":115},"ViewAction",{"@type":117,"mainEntity":118},"FAQPage",[119,125,129],{"name":120,"@type":121,"acceptedAnswer":122},"What is the purpose of an identity matrix in geometric transformations?","Question",{"text":123,"@type":124},"The identity matrix leaves coordinates unchanged, ensuring that 'right remains right' and 'up remains up' in the transformation process.","Answer",{"name":126,"@type":121,"acceptedAnswer":127},"How are rotations achieved using matrices?",{"text":128,"@type":124},"Rotations are performed by specific matrices, such as those for 90-degree clockwise, 90-degree anticlockwise, or 180-degree rotations, which map coordinates to new positions based on the rotation angle.",{"name":130,"@type":121,"acceptedAnswer":131},"What is the effect of an enlargement matrix with scale factors a and b?",{"text":132,"@type":124},"An enlargement matrix with scale factors a and b scales the x-coordinate by a factor of a and the y-coordinate by a factor of b, effectively stretching or shrinking the object along the respective axes.","https://schema.org",{"og:url":94,"og:type":135,"og:title":59,"og:site_name":106,"og:description":60},"article",{"robots":137,"canonical":94},"index,follow",{"doc_id":55,"site_id":69},1792086206]