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Question: this question tests if you can do basic linear algebra...

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This question tests if you can do basic linear algebra over arbitrary fields. Let Fs Z/52 be the finite field with 5 elements and consider the ring R F5[x]/(x3 +x + 1). We write α for the image of z in R. (a) Note that if β E R and c E F5, then cfe R. This gives us a scalar multiplication on R. Show that together with the normal addition on R, this makes R into a vector space over Fs (b) Multiplication by α gives a map μ: R R defined by μ(8) ad. Show that this map is Fs-linear. (c) Show that B = {1, α, α?) is an F5-basis for R. (d) Via part (b) we see that R-(vo + vie+U2αΑυο, ui, ½ E Fs). In particular, we have μ( 1) mo,o + moja + m0202, We can think of R as a column vector space via U2 We know that linear maps on vector spaces are given by matrix-vector multiplication. Express the map μ as a matrix-vector multiplication and write down the matrix explicitly. (e) Compute the characteristic polynomial of the matrix you computed in part (d).
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