transformations-using-a-matrix

Transformations using a matrix

What is a transformation matrix?

• A transformation matrix is used to determine the coordinates of an image from the transformation of an object

  • reflections, rotations, enlargements and stretches

  • Commonly used transformation matrices include

• (In 2D) a multiplication by any 2×2 matrix could be considered a transformation (in the 2D plane)

  • This can be done similarly in higher dimensions

• An individual point in the plane can be represented as a position vector, 

  • Several points, that create a shape say, can be written as a position matrix 

• A matrix transformation will be of the form <img alt=”open parentheses table row a b row c d end table close parentheses open parentheses table row x row y end table close parentheses” data-mathml='<math ><semantics><mrow><mfenced><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced></mrow><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“language”:”en”,”fontFamily”:”Times New Roman”,”fontSize”:”18″}</annotation></semantics></math>’ height=”54″ role=”math” src=”data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2254%22%20width%3D%2282%22%20wrs%3Abaseline%3D%2233%22%3E%3C!–MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmfenced%3E%3Cmtable%3E%3Cmtr%3E%3Cmtd%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmtd%3E%3Cmtd%3E%3Cmi%3Eb%3C%2Fmi%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3Cmtr%3E%3Cmtd%3E%3Cmi%3Ec%3C%2Fmi%3E%3C%2Fmtd%3E%3Cmtd%3E%3Cmi%3Ed%3C%2Fmi%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3C%2Fmtable%3E%3C%2Fmfenced%3E%3Cmfenced%3E%3Cmtable%3E%3Cmtr%3E%3Cmtd%3E%3Cmi%3Ex%3C%2Fmi%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3Cmtr%3E%3Cmtd%3E%3Cmi%3Ey%3C%2Fmi%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3C%2Fmtable%3E%3C%2Fmfenced%3E%3C%2Fmath%3E–%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A’brack_sm2882ad605b1e27be87c7468’%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7PH4UAAADMAAAATmNtYXA3kjw6AAABHAAAADxjdnQgAQYDiAAAAVgAAAASZ2x5ZkyYQ7YAAAFsAAAAkWhlYWQLyR8fAAACAAAAADZoaGVhAq0XCAAAAjgAAAAkaG10eDEjA%2FUAAAJcAAAADGxvY2EAAEKZAAACaAAAABBtYXhwBJsEcQAAAngAAAAgbmFtZW7QvZAAAAKYAAAB5XBvc3QArQBVAAAEgAAAACBwcmVwu5WEAAAABKAAAAAHAAACDAGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg9AMD%2FP%2F8AAABVAABAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACgAAAAGAAQAAQACI5wjn%2F%2F%2FAAAjnCOf%2F%2F%2FcZdxjAAEAAAAAAAAAAAFUAFQBAAArAIwAgACoAAcAAAACAAAAAADVAQEAAwAHAAAxMxEjFyM1M9XVq4CAAQHWqwABAAAAAABVAVgAAwAfGAGwAy%2BwADyxAgL1sAE8ALEDAD%2BwAjx8sQAG9bABPBEzESNVVQFY%2FqgAAQDXAAABLAFUAAMAIBgBsAUvsAE8sAI8sQAC9bADPACwAy%2BwAjyxAAH1sAE8EzMRI9dVVQFU%2FqwAAAAAAQAAAAEAAIsesexfDzz1AAMEAP%2F%2F%2F%2F%2FVre5k%2F%2F%2F%2F%2F9Wt7mT%2FgP%2F%2FAdYBWAAAAAoAAgABAAAAAAABAAABVP%2F%2FAAAXcP%2BA%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%2F8AAo2FAA%3D%3D)format(‘truetype’)%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%40font-face%7Bfont-family%3A’bracketse552f5417ff4