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Now let S : R4R³, with S(x, y, w, z) = (x+ w, -x +y-z, y-z+w) and T: R³ → R², with T(x, y, z)= (x - 2y +z, -2x + 4y - 2z). Find a matrix representation of the maps S and T with respect to the standard bases on R. Using these, show that the matrix of To S is given by ATOS = 3 -1 1 2 -6 2 -2 -4 Find ker(S) and ker(To S). Find the nullity of S and To S, denoted n(S) and n(TOS). State the rank-nullity theorem and use it to find the rank of S and To S, denoted r(S) and r(To S).

Now let S : R4R³, with S(x, y, w, z) = (x+ w, -x +y-z, y-z+w) and T: R³ → R², with T(x, y, z)= (x - 2y +z, -2x + 4y - 2z). Find a matrix representation of the maps S and T with respect to the standard bases on R. Using these, show that the matrix of To S is given by ATOS = 3 -1 1 2 -6 2 -2 -4 Find ker(S) and ker(To S). Find the nullity of S and To S, denoted n(S) and n(TOS). State the rank-nullity theorem and use it to find the rank of S and To S, denoted r(S) and r(To S).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Now let S : R4 →→ R³, with S(x, y, w,z) = (x +w, -x +y-z, y - z+w) and T : R³ → R², with T(x, y, z) = (x - 2y + z, −2x + 4y – 2z). Find a matrix representation of the maps S and T with respect to the standard bases on Rn. Using these, show that the matrix of To S is given by ATOS = -1 1 3 -6 2 -2 2 - Find ker(S) and ker (To S). Find the nullity of S and To S, denoted n(S) and n(To S). State the rank-nullity theorem and use it to find the rank of S and To S, denoted r(S) and r(To S).
Transcribed Image Text:Now let S : R4 →→ R³, with S(x, y, w,z) = (x +w, -x +y-z, y - z+w) and T : R³ → R², with T(x, y, z) = (x - 2y + z, −2x + 4y – 2z). Find a matrix representation of the maps S and T with respect to the standard bases on Rn. Using these, show that the matrix of To S is given by ATOS = -1 1 3 -6 2 -2 2 - Find ker(S) and ker (To S). Find the nullity of S and To S, denoted n(S) and n(To S). State the rank-nullity theorem and use it to find the rank of S and To S, denoted r(S) and r(To S).
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