By Kenneth L. Kuttler

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Thus for each x ∈ Rn , T x ∈ Rm . Then T is a linear transformation if whenever α, β are scalars and x1 and x2 are vectors in Rn , T (αx1 + βx2 ) = α1 T x1 + βT x2 . In words, linear transformations distribute across + and allow you to factor out scalars. At this point, recall the properties of matrix multiplication. 14 on Page 37. Recall it states that for a and b scalars, A (aB + bC) = aAB + bAC In particular, for A an m × n matrix and B and C, n × 1 matrices (column vectors) the above formula holds which is nothing more than the statement that matrix multiplication gives an example of a linear transformation.

7. Let A = . Is it possible to choose k such that AB = . Is it possible to choose k such that AB = 8. Let x = (−1, −1, 1) and y = (0, 1, 2) . Find xT y and xyT if possible. 9. Find the matrix for the linear transformation which rotates every vector in R2 through an angle of π/4. 10. Find the matrix for the linear transformation which rotates every vector in R2 through an angle of −π/3. 11. Find the matrix for the linear transformation which rotates every vector in R2 through an angle of 2π/3. 12.

1. BASIC TECHNIQUES AND PROPERTIES 59 First 1 2 −3 2 3 2 4 1 = det (AB) = det 11 −1 4 −4 AB = 11 −1 4 −4 and so = −40. Now det (A) = det 1 2 −3 2 det (B) = det 3 4 =8 and 2 1 = −5. Thus det (A) det (B) = 8 × (−5) = −40. 25 can be used to find determinants using row operations. As pointed out above, the method of Laplace expansion will not be practical for any matrix of large size. Here is an example in which all the row operations are used. 28 Find the determinant of the matrix, 1 2 3 4 5 1 2 3 A= 4 5 4 3 2 2 −4 5 Replace the second row by (−5) times the first row added to it.

### Multivariable Calculus, Applications and Theory [Lecture notes] by Kenneth L. Kuttler

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