Introduction to linear Algebra 4th. Edition by Gilbert Strang | 9780980232714 | 0980232716 | Holooly

Introduction to linear Algebra [EXP-672]

1062 SOLVED PROBLEMS

Question: 2.1.16

( a) What 2 by 2 matrix R rotates every vector by 90°? R times [ x y ] is [ y -x ]. (b) What 2 by 2 matrix R^2 rotates every vector by 180°?

Verified Answer:

( a) 90° rotation from

R=\left[ \begin{mat...

Question: 2.4.11

(3 by 3 matrices) Choose the only B so that for every matrix A (a) BA = 4A (b) BA = 4B (c) BA has rows 1 and 3 of A reversed and row 2 unchanged (d) All rows of BA are the same as row 1 of A.

Verified Answer:

(a) B=4I (b) B=0 (c)

B=\left[ \begin{matrix...

(A calculus question) Show that the derivative of det A with respect to a11 is the cofactor C11. The other entries are fixed-we are only changing a11.

Verified Answer:

(a) det A =

{a}_{11}{C}_{11}+ \cdots + {a}_...

Question: 6.4.22

(A paradox for instructors) If AA^T = A^T A then A and A^T share the same eigenvectors (true). A and A^T always share the same eigenvalues. Find the flaw in this conclusion: They must have the same S and Λ. Therefore A equals A^T.

Verified Answer:

A and

{A}^{T}

have the same

...

(a) Check that the solutions to Ax = 0 are perpendicular to the rows: A = [ 1 0 0 2 1 0 3 4 1 ] [ 4 2 0 1 0 0 1 3 0 0 0 0 ]=ER. (b) How many independent solutions to A^T y = 0? Why is y^T the last row of E^-1?

Verified Answer:

(a) Special solutions (-1, 2, 0, 0) and (

-\...

(a) Choose c so that Q is an orthogonal matrix: Q=c [ 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 ] . Project b = (1, 1, 1, 1) onto the first column. Then project b onto the plane of the first two columns.

Verified Answer:

(a)

c = \frac{1}{2}

normalizes all ...

(a) Choose E21 to remove the 3 below the first pivot. Then multiply E21 A E^T21 to remove both 3 ‘s: A= [ 1 3 0 3 11 4 0 4 9 ] is going toward D = [ 1 0 0 0 2 0 0 0 1 ]. (b) Choose E32 to remove the 4 below the second pivot. Then A is reduced to D by E32 E21 A E^T21 E^T32 = D. Invert the E’s to

Verified Answer:

(a)

{E}_{21}=\left[ \begin{matrix} 1 & ...

(a) Choose sinθ and cosθ to triangularize A, and find R: Givens rotation Q21 A = [ cosθ -sinθ sinθ cosθ ] [ 1 -1 3 5 ]= [ * * 0 * ]=R. (b) Choose sinθ and cosθ to make QAQ^-1 triangular. What are the eigenvalues?

Verified Answer:

(a)

\cos {\theta} = 1/\sqrt{10}

, [l...

(a) Compute A^TA and its eigenvalues and unit eigenvectors v1 and v2. Find σ1. Rank one matrix A =[ 1 2 3 6 ] (b) Compute AA^T and its eigenvalues and unit eigenvectors u1 and u2. (c) Verify that Av1= σ1u1. Put numbers into the SVD: A=UΣV^T [ 1 2 3 6 ] = [ u1 u2 ] [ σ1 0 ] [ v1 v2 ]^T.

Verified Answer:

{A}^{T}A =\begin{bmatrix} 10 & 20 \\ 20...

(a) Compute P = QQ^T when q1 = (.8, .6,0) and q2 = (-.6, .8,0). Verify that P^2 = P. (b) Prove that always (QQ^T)^2 = QQ^T by using Q^T Q = I. Then P = QQ^T is the projection matrix onto the column space of Q.

Verified Answer:

(a)

Q =\left[ \begin{matrix} .8 & -.6 \...

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