In Exercises 9-12, write the system of linear equations represented by the augmented matrix. Use x , y, and or, if necessary, w , x, y. and z, for the variables. [ 4 1 5 1 1 − 1 0 − 1 3 0 0 7 0 0 11 7 | 6 8 4 3 ]
In Exercises 9-12, write the system of linear equations represented by the augmented matrix. Use x , y, and or, if necessary, w , x, y. and z, for the variables. [ 4 1 5 1 1 − 1 0 − 1 3 0 0 7 0 0 11 7 | 6 8 4 3 ]
Solution Summary: The author explains how a system of linear equations can be solved with the help of matrices.
In Exercises 9-12, write the system of linear equations represented by the augmented matrix. Use x, y, and or, if necessary, w, x, y. and z, for the variables.
R denotes the field of real numbers, Q denotes the field of rationals, and
Fp denotes the field of p elements given by integers modulo p. You may refer to general
results from lectures.
Question 1
For each non-negative integer m, let R[x]m denote the
vector space consisting of the polynomials in x with coefficients in R and of degree ≤ m.
x²+2, V3 = 5. Prove that (V1, V2, V3) is a linearly independent
(a) Let vi = x, V2 =
list in R[x] 3.
(b) Let V1, V2, V3 be as defined in (a). Find a vector v € R[×]3 such that (V1, V2, V3, V4)
is a basis of R[x] 3.
[8]
[6]
(c) Prove that the map ƒ from R[x] 2 to R[x]3 given by f(p(x)) = xp(x) — xp(0) is a
linear map.
[6]
(d) Write down the matrix for the map ƒ defined in (c) with respect to the basis
(2,2x + 1, x²) of R[x] 2 and the basis (1, x, x², x³) of R[x] 3.
[5]
Question 4
(a) The following matrices represent linear maps on R² with respect to an
orthonormal basis:
=
[1/√5 2/√5
[2/√5 -1/√5]
"
[1/√5 2/√5]
A =
B =
[2/√5 1/√5] 1
C =
D =
=
=
[ 1/3/5 2/35]
1/√5 2/√5
-2/√5 1/√5'
For each of the matrices A, B, C, D, state whether it represents a self-adjoint
linear map, an orthogonal linear map, both, or neither.
(b) For the quadratic form
q(x, y, z) = y² + 2xy +2yz
over R, write down a linear change of variables to u, v, w such that q in these
terms is in canonical form for Sylvester's Law of Inertia.
[6]
[4]
part b please
Chapter 9 Solutions
MyLab Math with Pearson eText -- Combo Access Card (18-wk) for Algebra & Trigonometry
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