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Chapter 1: Q55E (page 1)
If Ais a matrix with eigenvalues 3 and 4 and if localid="1668109698541" is a unit eigenvector of A, then the length of vector Alocalid="1668109419151" cannot exceed 4.
The given statement is True because the given matrices are not exceeded 4.
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Suppose \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2}} \right\}\) is a linearly independent set in \({\mathbb{R}^n}\). Show that \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm v}\nolimits} _2}} \right\}\) is also linearly independent.
Solve the linear system of equations. You may use technology.
Construct a \(2 \times 3\) matrix \(A\), not in echelon form, such that the solution of \(Ax = 0\) is a line in \({\mathbb{R}^3}\).
The solutions \(\left( {x,y,z} \right)\) of a single linear equation \(ax + by + cz = d\)
form a plane in \({\mathbb{R}^3}\) when a, b, and c are not all zero. Construct sets of three linear equations whose graphs (a) intersect in a single line, (b) intersect in a single point, and (c) have no points in common. Typical graphs are illustrated in the figure.
Three planes intersecting in a line.
(a)
Three planes intersecting in a point.
(b)
Three planes with no intersection.
(c)
Three planes with no intersection.
(肠鈥�)
In Exercises 23 and 24, key statements from this section are either quoted directly, restated slightly (but still true), or altered in some way that makes them false in some cases. Mark each statement True or False, and justify your answer. (If true, give the approximate location where a similar statement appears, or refer to a de铿乶ition or theorem. If false, give the location of a statement that has been quoted or used incorrectly, or cite an example that shows the statement is not true in all cases.) Similar true/false questions will appear in many sections of the text.
23.
a. Every elementary row operation is reversible.
b. A \(5 \times 6\)matrix has six rows.
c. The solution set of a linear system involving variables \({x_1},\,{x_2},\,{x_3},........,{x_n}\)is a list of numbers \(\left( {{s_1},\, {s_2},\,{s_3},........,{s_n}} \right)\) that makes each equation in the system a true statement when the values \ ({s_1},\, {s_2},\, {s_3},........,{s_n}\) are substituted for \({x_1},\,{x_2},\,{x_3},........,{x_n}\), respectively.
d. Two fundamental questions about a linear system involve existence and uniqueness.
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