Introduction to Linear Algebra 4th Edition solutions

Introduction to Linear Algebra 4
A linear transformation must leave the zero vector fixed: T(O) = O. Prove this from T(v + w) = T(v) + T(w) by choosing w = (and finish the proof). Prove it also from T(cv) = cT(v) by choosing c = __

Introduction to Linear Algebra 4
Requirement (b) gives T(cv) = cT(v) and also T(dw) = dT(w). Then by addition, requirement (a) gives T( ) = ( ). What is T(cv + dw + eu)?

Introduction to Linear Algebra 4
Which of these transformations are not linear? The input is v = (VI, V2): (a) T(v) = (V2, vd (d) T(v) = (0,1) (b) T(v) = (VI, vd (e) T(v) = VI - V2 (c) T(v) = (0, vd (f) T(v) = VI V2

Introduction to Linear Algebra 4
If Sand T are linear transformations, is S (T (v linear or quadratic? (a) (Special case) If S(v) = v and T(v) = v, then S(T(v) = v or v 2? (b) (General case) S(WI +W2) = S(WI)+S(W2) and T(VI +V2) = T(vd+T(V2) com bine into S(T(VI + V2 = S( __ ) = __ + __

Introduction to Linear Algebra 4
Suppose T(v) = v except that T(O, V2) = (0,0). Show that this transformation satisfies T(cv) = cT(v) but not T(v + w) = T(v) + T(w).

Introduction to Linear Algebra 4
Which of these transformations satisfy T (v + w) = T (v) + T ( w) and which satisfy T(cv) = cT(v)? (a) T(v) = v/llvll (d) T(v) = largest component of v.

Introduction to Linear Algebra 4
For these transformations of V = R 2 to W = R 2 , find T (T (v. Is this transformation T2 linear? (a) T(v) =-v (b) T(v) = v + (1, 1) (c) T(v) = 900 rotation = (-V2, vd (d) T(v) = projection = (VI ~V2, VI ~V2).

Introduction to Linear Algebra 4
Find the range and kernel (like the column space and nullspace) of T: (a) T(v!, V2) = (VI - V2, 0) (c) T(VI' V2) = (0,0) (b) T(VI' V2, V3) = (VI, V2) (d) T(VI' V2) = (VI, VI).

Introduction to Linear Algebra 4
The "cyclic" transformation T is defined by T(Vl' V2, V3) = (V2' V3, vd. What is T(T(v? What is T 3 (v)? What is TIOO(V)? Apply T a hundred times to v.

Introduction to Linear Algebra 4
A linear transformation from V to W has an inverse from W to V when the range is all ofW and the kernel contains only v = O. Then T(v) = w has one solution v for each w in W. Why are these T's not invertible? (a) T(VI' V2) = (V2, V2) (b) T(VI' V2) = (VI, V2, VI + V2) (c) T(VI' V2) = V

Introduction to Linear Algebra 4
If T(v) = Av and A is m by n, then T is "multiplication by A." (a) What are the input and output spaces V and W? (b) Why is range of T = column space of A? ( c) Why is kernel of T = nullspace of A?

Introduction to Linear Algebra 4
Suppose a linear T transforms (1, 1) to (2,2) and (2,0) to (0,0). Find T(v): (a) v = (2,2) (b) v = (3,1) (c) v=(-1,1) (d) v = (a, b).

Introduction to Linear Algebra 4
Problems 13-19 may be harder. The input space V contains all 2 by 2 matrices M.

Introduction to Linear Algebra 4
Problems 13-19 may be harder. The input space V contains all 2 by 2 matrices M.

Introduction to Linear Algebra 4
Problems 13-19 may be harder. The input space V contains all 2 by 2 matrices M.

Introduction to Linear Algebra 4
Problems 13-19 may be harder. The input space V contains all 2 by 2 matrices M.

Introduction to Linear Algebra 4
Problems 13-19 may be harder. The input space V contains all 2 by 2 matrices M.

Introduction to Linear Algebra 4
Problems 13-19 may be harder. The input space V contains all 2 by 2 matrices M.

Introduction to Linear Algebra 4
Problems 13-19 may be harder. The input space V contains all 2 by 2 matrices M.

Introduction to Linear Algebra 4
Questions 20-26 are about house transformations. The output is T(H) = A H.

Introduction to Linear Algebra 4
Questions 20-26 are about house transformations. The output is T(H) = A H.

Introduction to Linear Algebra 4
Questions 20-26 are about house transformations. The output is T(H) = A H.

Introduction to Linear Algebra 4
Questions 20-26 are about house transformations. The output is T(H) = A H.

Introduction to Linear Algebra 4
Questions 20-26 are about house transformations. The output is T(H) = A H.

Introduction to Linear Algebra 4
Questions 20-26 are about house transformations. The output is T(H) = A H.

Introduction to Linear Algebra 4
Questions 20-26 are about house transformations. The output is T(H) = A H.

Introduction to Linear Algebra 4
This code creates a vector theta of 50 angles. It draws the unit circle and then T (circle) = ellipse. T(v) = Av takes circles to ellipses. A = [21;1 2] % You can change A theta = [0:2 * pi/SO:2 * pi]; circle = [cos(theta); sin(theta)]; ellipse = A * circle; axis([-4 4 -44]); axis('square') plot(circ

Introduction to Linear Algebra 4
Add two eyes and a smile to the circle in Problem 27. (If one eye is dark and the other is light, you can tell when the face is reflected across the y axis.) Multiply by matrices A to get new faces.

Introduction to Linear Algebra 4
What conditions on det A = ad - be ensure that the output house AH will (a) be squashed onto a line? (b) keep its endpoints in clockwise order (not reflected)? (c) have the same area as the original house?

Introduction to Linear Algebra 4
From A = U 1: VT (Singular Value Decomposition) A takes circles to ellipses. A V = U 1: says that the radius vectors VI and V2 of the circle go to the semi-axes atUl and a2u2 of the ellipse. Draw the circle and the ellipse for e = 30: V=[~ b] U = [ C?S e - si

Introduction to Linear Algebra 4
Why does every linear transformation T from R2 to R2 take squares to parallelograms? Rectangles also go to parallelograms (squashed if T is not invertible).