Linear

Linear Algebra, Theory And Applications by Kenneth L. Kuttler

By Kenneth L. Kuttler

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The main diagonal consists of every entry of the matrix which is of the form aii . It runs from the upper left down to the lower right. 4. 8) show −A is unique. 5. 8) show 0 is unique. 6. 8) show 0A = 0. Here the 0 on the left is the scalar 0 and the 0 on the right is the zero for m × n matrices. 7. 8) and previous problems show (−1) A = −A. 8. 17). 9. Prove that Im A = A where A is an m × n matrix. 2. EXERCISES 55 n m 10. Let ( AT and ) be a real m × n matrix and let x ∈ R and y ∈ R . Show (Ax, y)Rm = x,A y Rn where (·, ·)Rk denotes the dot product in Rk .

V =  .  , A = (a1 , · · · , an ) vn where ai is an m × 1 vector. Then Av, written as  ( ) a1 · · · an    v1 ..   . , vn is the m × 1 column vector which equals the following linear combination of the columns. 9) 42 CHAPTER 2. LINEAR TRANSFORMATIONS If the j th column of A is       A1j A2j .. 9) takes the form    v1    A11 A21 .. Am1        + v2      A12 A22 .. Am2        + · · · + vn      A1n A2n .. Amn       ∑n Thus the ith entry of Av is j=1 Aij vj .

Ei ≡  1  ,    ..   .  0 where the 1 is in the ith position and there are zeros everywhere else. Thus T ei = (0, · · · , 0, 1, 0, · · · , 0) . Of course the ei for a particular value of i in Fn would be diﬀerent than the ei for that same value of i in Fm for m ̸= n. One of them is longer than the other. However, which one is meant will be determined by the context in which they occur. These vectors have a signiﬁcant property. 3 Let v ∈ Fn . Thus v is a list of numbers arranged vertically, v1 , · · · , vn .