Answer to Question #235293 in Linear Algebra for Precious

Question #235293

Find the standard matrix A for the linearly transformation T :R^2_R^2


1
Expert's answer
2021-09-10T10:03:35-0400

Find the standard matrix of "T:\\R^2\\to \\R^2" rotates points (about the origin) through "\u2212\u03c0\/4" radians (clockwise).

Let "T:\\R^2\\to \\R^2" be a linear transformation given by rotating vectors in the counter-clockwise direction through an angle of "\u03b8."

Then the matrix "A" of "T" is given by


"T\\bigg(\\begin{pmatrix}\n 1 \\\\\n 0\n\\end{pmatrix}\\bigg)=\\begin{pmatrix}\n \\cos \\theta \\\\\n\\sin \\theta\n\\end{pmatrix}"

"T\\bigg(\\begin{pmatrix}\n 0 \\\\\n 1\n\\end{pmatrix}\\bigg)=\\begin{pmatrix}\n - \\sin \\theta \\\\\n\\cos \\theta\n\\end{pmatrix}"

"A=\\begin{pmatrix}\n \\cos \\theta & - \\sin\\theta \\\\\n \\sin \\theta & \\cos \\theta\n\\end{pmatrix}"

Given "\\theta=\\pi\/4." Then


"A=\\begin{pmatrix}\n \\cos (\\pi\/4) & - \\sin (\\pi\/4) \\\\\n \\sin (\\pi\/4) & \\cos(\\pi\/4)\n\\end{pmatrix}=\\begin{pmatrix}\n \\sqrt{2}\/2& -\\sqrt{2}\/2 \\\\\n \\sqrt{2}\/2 & \\sqrt{2}\/2\n\\end{pmatrix}"


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