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Unformatted text preview: ∈ R n × n , is the equation ( A + B ) 2 = A 2 +2 AB + B 2 still true? If so, prove it, if not, ﬁnd examples of matrices A and B for which this fails. 4. Let A = 2 4 8 4 5 1 7 9 3 . (a) Find ker A and im A as spans of vectors. 3 (b) Find all solutions of the system A x = b , where b = 6 9 16 . 5. Consider the basis β = ±² 1 1 ³ , ² 1 2 ³´ for R 2 . Find the representation [ x ] β of the vector x = ²2 5 ³ in this basis. 6. Let ρ = { e 1 , e 2 } be the standard basis for R 2 and let β = ±² 1 1 ³ , ² 1 ³´ be another basis for R 2 . If T A ∈ L ( R 2 , R 2 ) is the counterclockwise rotation through π/ 4, with associated matrix A = " 1 √ 21 √ 2 1 √ 2 1 √ 2 # ﬁnd the matrix representation [ T A ] ρ,β of T A in these bases. 4...
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 Spring '12
 DavidGlenn
 Linear Algebra, Algebra, Equations, Vectors, basis, TA, Euclidean vector

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