Questions and Answers of Linear Algebra And Its Applications

Mark each statement True or False (T/F). Justify each answer.(T/F) The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process.
Mark each statement True or False (T/F). Justify each answer.(T/F) The solution set of the linear system whose augmented matrix is [a₁ a2 a3 b] is the same as the solution set of the equation x₁
Mark each statement True or False (T/F). Justify each answer.(T/F) The first entry in the product Ax is a sum of products.
Note thatUse this fact (and no row operations) to find scalars c1, c2, c3 such that [2 3-4 6-3 2 4 -8 9 9-5 G]-[ 3 -1 -2 -9 8
Mark each statement True or False (T/F). Justify each answer.(T/F) When u and v are nonzero vectors, Span {u, v} contains the line through u and the origin.
LetIt can be shown that 3u - 5v - w = 0. Use this fact (and no row operations) to find x1 and x2 that satisfy the equation U= 7 2 5 V= 3 1 3 and w= 6 [1] 0
Mark each statement True or False (T/F). Justify each answer.(T/F) Any linear combination vectors can always be written in the form Ax for a suitable matrix A and vector x.
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
An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin plate when the temperature around the boundary is known. Assume the plate shown
Suppose the system below is consistent for all possible values of f and g. What can you say about the coefficients c and d? Justify your answer. x₁ + 5x₂ = f cx1 + dx₂ = g
Mark each statement True or False (T/F). Justify each answer.(T/F) Reducing a matrix to echelon form is called the forward phase of the row reduction process.
Mark each statement True or False (T/F). Justify each answer.(T/F) The set Span {u, v} is always visualized as a plane through the origin.
Mark each statement True or False (T/F). Justify each answer.(T/F) If the columns of an m x n matrix A span RM, then theequation Ax = b is consistent for each b in RM.
Mark each statement True or False (T/F). Justify each answer.(T/F) Finding a parametric description of the solution set of a linear system is the same as solving the system.
Rewrite the (numerical) matrix equation below in symbolic form as a vector equation, using symbols V₁, V2,... for the vectors and C₁, C2,... for scalars. Define what each symbol represents, using
Mark each statement True or False (T/F). Justify each answer.(T/F) Asking whether the linear system corresponding to an augmented matrix [a₁ a2 a3 b] has a solution amounts to asking whether b is
Mark each statement True or False (T/F). Justify each answer.(T/F) The solution set of a linear system whose augmented matrix is [a₁ a₂ a3 b] is the same as the solution set of Ax = b, if A =
Consider the vectors V₁, V2, V3, and b in R2, shown in thefigure. Does the equation X₁V₁ + X2V2 + x3V3 = b have a solution? Is the solution unique? Use the figure to explain your answers.
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
Mark each statement True or False (T/F). Justify each answer.(T/F) Whenever a system has free variables, the solution set contains a unique solution.
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
Mark each statement True or False (T/F). Justify each answer.(T/F) If one row in an echelon form of an augmented matrix is [0 0 0 0 5], then the associated linear system is inconsistent.
Mark each statement True or False (T/F). Justify each answer.(T/F) If the augmented matrix [ A b] has a pivot position in every row, then the equation Ax = b is inconsistent.
Construct three different augmented matrices for linear systems whose solution set is x₁ = -2, x₂ = 1, x3 = 0.
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
Mark each statement True or False (T/F). Justify each answer.(T/F) A general solution of a system is an explicit description of all solutions of the system.
Find the interpolating polynomial p(t) = a0 + a₁t + a₂t² for the data (1, 11), (2, 16), (3, 19). That is, find a0, a1, anda2 such that ao + a1(1) + a₂(1)² = 11 ao + a1(2) + a₂ (2)² = 16 ao
Suppose the coefficient matrix of a linear system of three equations in three variables has a pivot in each column. Explain why the system has a unique solution.
Determine if the columns of the matrix span R4. 5 6 4 -9 -7 -8 -4 11 -4 -7 9 5 -9 -9 16 7
Determine if the columns of the matrix span R4. 7 -5 2-5 56 -3 6 10 -2 54 -7 9 NN 2 8 -9 7 15
Give an example of an inconsistent underdetermined system of two equations in three unknowns.
What would you have to know about the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution?
Suppose A is a 3 x 3 matrix and b is a vector in R³ with the property that Ax = b has a unique solution. Explain why the columns of A must span R³.
Use the vectors u = (u₁,...,Un), v = (v₁,..., vn), and w = (ω₁,..., ωn) to verify the following algebraic proper-ties of Rn.a. (u + v) + w = u+ (v + w)b. c(u + v) = cu + cv for each scalar c
Use the vector u = (u₁,..., un) to verify the following algebraic properties of Rn.a. u + (-u) = (-u) + u = 0b. c(du) (cd)u for all scalars c and d
Suppose an n x (n + 1) matrix is row reduced to reduced echelon form. Approximately what fraction of the total num- ber of operations (flops) is involved in the backward phase of the reduction when n
Let A be a 3 x 4 matrix, let y₁ and y₂ be vectors in R³, and let w = y₁ + y₂. Suppose y₁ = Ax₁ and y₂ = Ax₂ for some vectors x₁ and x2 in R4. What fact allows you to concludethat
Determine if the columns of the matrix span R4. 12 -7 -9 4 -6 11 4 -6 11 -9 -8 7 -7 3 10 -5 5 -3 -9 12
Determine if the columns of the matrix span R4. 8 11 -6 -7 -7 -8 5 6 11 7 -7 -9 -3 4 4 18 13 -9 -6 7
Consider each matrix as the augmented matrix of a linear system. State in words the next two elementary row operations that should be performed in the process of solving the system. 1-6 4 2
Find the point of intersection of the lines x₁ - 5x₂ = 1 and 3x1 - 7x₂ = 5.
Find the point (x1, x2) that lies on the line x? + 5x2 = 7 and on the line x? - 2x2 = -2. See the figure. *ị+ 5x = 7 x2 x₁ - 2x₂ = -2 XI
Using elementary row operations on the equations or on the augmented matrix. Follow the systematic elimination procedure described in this section. 2x1 + 4x₂ = −4 -4 5x17x₂= 11 ||

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