Answer to Question #270204 in Linear Algebra for Gestavo

Question #270204

Determine if the following transformation is a linear transformation:

T (x1, x2) → (cos(x1), sin(x2))

T = R2→R2



1
Expert's answer
2021-11-23T11:50:20-0500

"\\displaystyle\\text{Given } T\\binom{x_1}{x_2} = \\binom{\\cos x_1}{\\sin x_2} \\\\\n\\text{Let $\\alpha$ be a scalar and }\\binom{y_1}{y_2} \\text{ be vector then: } \\\\\nT\\left(\\binom{x_1 }{x_2} + \\alpha \\binom{y_1}{y_2}\\right) = T\\binom{x_1+ \\alpha y_1}{x_2 + \\alpha y_2} \\\\\n\\qquad \\qquad \\qquad \\qquad\\quad= \\binom{\\cos(x_1 + \\alpha y_1)}{\\sin(x_2+ \\alpha y_2)} = \\binom{\\cos x_1\\cos \\alpha y_1 - \\sin x_1\\sin \\alpha y_1}{\\sin x_1\\cos \\alpha y_1 + \\cos x_1\\sin \\alpha y_1} \\\\\n\\text{And } \\\\\nT\\binom{x_1 }{x_2} + \\alpha\\, T\\binom{y_1}{y_2} = \\binom{\\cos x_1}{\\sin x_2} + \\alpha \\, \\binom{\\cos y_1}{\\sin y_2} = \\binom{\\cos x_1 +\\alpha \\cos y_1}{\\sin x_2 + \\alpha \\sin y_2} \\ne \\binom{\\cos x_1\\cos \\alpha y_1 - \\sin x_1\\sin \\alpha y_1}{\\sin x_1\\cos \\alpha y_1 + \\cos x_1\\sin \\alpha y_1} = T\\left(\\binom{x_1 }{x_2} + \\alpha \\binom{y_1}{y_2}\\right) \\\\\n\\implies T\\binom{x_1 }{x_2} + \\alpha\\, T\\binom{y_1}{y_2} \\ne T\\left(\\binom{x_1 }{x_2} + \\alpha \\binom{y_1}{y_2}\\right) \\\\\n\\therefore \\text{The given transformation is not a linear transformation.}"


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