(a) Let A € R²x2, Prove that (Write A(ax+By) =aAx+ Ay for all a, ß € R, x, y € R². ^-[22][3)--[G] ], x= [ 2₂ ]· y = [ 2 ] A = = all a12 921 and write out A(ax+By) explicitly.) Now repeat part 13(a) for A € R"x". The proof is not difficult when one uses summation notation. For example, if the (i, j)-entry of A is denoted a;;, then (Ax); = Σaijxj. j=1 Write (A(ax +By)); and (@Ax+ ßAy); in summation notation, and show that the first can be rewritten as the second using the elementary properties of arith- metic.
(a) Let A € R²x2, Prove that (Write A(ax+By) =aAx+ Ay for all a, ß € R, x, y € R². ^-[22][3)--[G] ], x= [ 2₂ ]· y = [ 2 ] A = = all a12 921 and write out A(ax+By) explicitly.) Now repeat part 13(a) for A € R"x". The proof is not difficult when one uses summation notation. For example, if the (i, j)-entry of A is denoted a;;, then (Ax); = Σaijxj. j=1 Write (A(ax +By)); and (@Ax+ ßAy); in summation notation, and show that the first can be rewritten as the second using the elementary properties of arith- metic.
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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![13. (a) Let A € R²×2. Prove that
(Write
A(ax+By) = αAx+ BAy for all a, ß € R, x, y € R².
=[ 22]×=[^]=[]
] ₁ x = [ 2 ] · y = [ 2 ]
X=
X2
all a12
a21
and write out A(ax+By) explicitly.)
(b) Now repeat part 13(a) for A € R"x". The proof is not difficult when one uses
summation notation. For example, if the (i, j)-entry of A is denoted a;;, then
(Ax); = Ĺaijxj.
j=1
Write (A(ax+By)); and (aAx+BAy); in summation notation, and show that
the first can be rewritten as the second using the elementary properties of arith-
metic.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc7559a47-bb34-494d-9e33-22c98791555e%2F2b1a686b-d9e6-4c36-b220-640300b0aa8e%2Fu1jnop_processed.jpeg&w=3840&q=75)
Transcribed Image Text:13. (a) Let A € R²×2. Prove that
(Write
A(ax+By) = αAx+ BAy for all a, ß € R, x, y € R².
=[ 22]×=[^]=[]
] ₁ x = [ 2 ] · y = [ 2 ]
X=
X2
all a12
a21
and write out A(ax+By) explicitly.)
(b) Now repeat part 13(a) for A € R"x". The proof is not difficult when one uses
summation notation. For example, if the (i, j)-entry of A is denoted a;;, then
(Ax); = Ĺaijxj.
j=1
Write (A(ax+By)); and (aAx+BAy); in summation notation, and show that
the first can be rewritten as the second using the elementary properties of arith-
metic.
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