Choose The One Alternative That Best Completes The Sentence

Choose The One Alternative That Best Completes The S

Multiple Choice Choose The One Alternative That Best Completes The S

Multiple Choice Choose The One Alternative That Best Completes The S

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1 Solve the system of equations. 1) x1 + x2 + x3 = 7 x1 - x2 + 2x3 = 7 5x1 + x2 + x3 = 11 A) (4, 1, 2) B) (1, 2, 4) C) (4, 2, 1) D) (1, 4, ) Determine whether the system is consistent. 2) x1 + x2 + x3 = 7 x1 - x2 + 2x3 = 7 5x1 + x2 + x3 = 11 A) No B) Yes 2) Determine whether the matrix is in echelon form, reduced echelon form, or neither. A) Echelon form B) Neither C) Reduced echelon form A) Echelon form B) Reduced echelon form C) Neither Find the indicated vector. 5) Let u = -2 -3 A) . Find -9 u . B) - C) -) D) ) Let u = -3 2 A) . Find 7 u . B) C) - D) -) Compute the product or state that it is undefined. 7) [-6 2 5] A) [-51] Write the system as a vector equation or matrix equation as indicated. 8) Write the following system as a vector equation involving a linear combination of vectors. 3x1 - 5x2 - x3 = 2 5x1 + 3x3 = ) A) x + x + x3 1 = x1 B) 3 x2 x3 3 x x2 - x3 5 x1 2 x2 = 6 x C) x + x + x = 2 D) x + x = Solve the problem. 9) Find the general solution of the homogeneous system below.

Give your answer as a vector. x1 + 2x2 - 3x3 = 0 4x1 + 7x2 - 9x3 = 0 -x1 - 3x2 + 6x3 = ) A) x1 -3 x2 = x3 3 x3 0 C) x1 3 B) x1 -3 x2 = x3 3 x3 1 D) x1 -3 x2 = x3 -3 x3 1 x2 = 3 x) Find the general solution of the simple homogeneous ʺsystemʺ below, which consists of a single linear equation. Give your answer as a linear combination of vectors. Let x2 and x3 be free variables. -2x1 - 14x2 + 8x3 = 0 A) 10) x1 -7 x2 = x2 0 x3 1 B) x1 x1 4 + x x1 (with x2, x3 free) x2 = -7 x2 x3 x3 C) x x2 x (with x2, x3 free) x2 = x2 x3 D) x1 x2 = x2 x3 1 + x + x (with x2, x3 free) (with x2, x3 free) 11) Suppose an economy consists of three sectors: Energy (E), Manufacturing (M), and Agriculture (A). Sector E sells 70% of its output to M and 30% to A.

Sector M sells 30% of its output to E, 50% to A, and retains the rest. Sector A sells 15% of its output to E, 30% to M, and retains the rest. Denote the prices (dollar values) of the total annual outputs of the Energy, Manufacturing, and Agriculture sectors by pe, pm, and pa, respectively. If possible, find equilibrium prices that make each sectorʹs income match its expenditures. Find the general solution as a vector, with pa free.

11) A) pe pm = pa C) pe pm = pa 0.308 pa 0.716 pa pa 0.607 pa 0.481 pa pa B) pe pm = pa D) pe pm = pa 0.465 pa 0.593 pa pa 0.356 pa 0.686 pa pa 12) The network in the figure shows the traffic flow (in vehicles per hour) over several one -way streets in the downtown area of a certain city during a typical lunch time. Determine the general flow pattern for the network. In other words, find the general solution of the system of equations that describes the flow. In your general solution let x4 be free. 13) For what values of h are the given vectors linearly independent? , 24 1 h A) Vectors are linearly independent for h = -4 B) Vectors are linearly dependent for all h C) Vectors are linearly independent for all h D) Vectors are linearly independent for h ≠-) Let v1 = -3 8 , v2 = -3 8 , v3 = . -) Determine if the set { v1, v2, v3 } is linearly independent.

A) Yes B) No Describe geometrically the effect of the transformation T. ) Let A = . Define a transformation T by T( x ) = A x . A) Projection onto the x2-axis B) Horizontal shear C) Projection onto the x2x3-plane D) Vertical shear Solve the problem. 16) Let T: â„›2 -> â„›2 be a linear transformation that maps u = -3 4 into -13 6 and maps v = 4 6 into . -8 Use the fact that T is linear to find the image of 3 u + v . A) B) - C) - D) - Determine whether the linear transformation T is one -to-one and whether it maps as specified.

17) T(x1, x2, x3) = (-2x2 - 2x3, -2x1 + 8x2 + 4x3, -x1 - 2x3, 4x2 + 4x3) Determine whether the linear transformation T is one -to-one and whether it maps â„›3 onto â„›4. A) One-to-one; onto â„›4 B) One-to-one; not onto â„›4 C) Not one-to-one; not onto â„›4 D) Not one-to-one; onto â„›) Let T be the linear transformation whose standard matrix is A = - . Determine whether the linear transformation T is one -to-one and whether it maps â„›3 onto â„›3. A) Not one-to-one; not onto â„›3 B) One-to-one; not onto â„›3 C) Not one-to-one; onto â„›3 D) One-to-one; onto â„›3 Find the matrix product AB, if it is defined. 19) A = , B = -1 3 2 . ) A) B) C) AB is undefined.

D) The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, if the products are defined. 20) A is 2 à— 1, B is 1 à— 1. A) AB is 1 à— 2, BA is 1 à— 1. B) AB is 2 à— 1, BA is undefined.

C) AB is undefined, BA is 1à— 2. D) AB is 2 à— 2, BA is 1 à— 1. 20) Solve the system by using the inverse of the coefficient matrix. 21) 10x1 - 4x2 = -6 6x1 - x2 = 2 A) (-4, -1) B) (4, 1) C) (1, 4) D) (-1, -x1 - 6x2 = -6 3x1 + 2x2 = 13 A) (3, 2) B) (2, 3) C) (-3, -2) D) (-2, -) Determine whether the matrix is invertible. 23) A) No B) Yes A) Yes B) No 24) Matrix A shown below is associated with a linear transformation and T (x)= Ax 1 3 A = .

Find 1 T T . Find 2 T T . Linear Transformation T is a mapping from R2 → R2 and if T is onto then for every b in R2 there is at least one x in R2 where T (x)= b . or said with matrices Ax = b has a solution for every possible b . Determine if T is "onto" or not. 4.

Linear Transformation T is a mapping from R2 → R2 and if T is one-to-one then the only value in R2 where T (x)= 0. is x = 0 or said with matrices Ax = 0 Determine if T is one-to-one or not. 5. Show that the following Linear Transformation is NOT linear x 1 − 2 x 2 x 1 T = x 1 − 3 x 2 2 x 1 − 5 x 2

Paper For Above instruction

This comprehensive analysis explores fundamental topics in linear algebra, including solving systems of equations, matrix forms, vector operations, linear independence, transformations, and matrix invertibility, all crucial for advanced mathematical understanding and applications.

Initial problem-solving involves systems of linear equations. For instance, solving a system with three equations and three variables typically involves using elimination, substitution, or matrix methods such as Gaussian elimination or matrix inverses. The specific system:

x₁ + x₂ + x₃ = 7

x₁ - x₂ + 2x₃ = 7

5x₁ + x₂ + x₃ = 11

can be solved using matrix form. The coefficient matrix and augmented matrix help determine the solution set, revealing whether solutions are unique, infinite, or nonexistent. Applying Gaussian elimination shows the system's consistency and solutions, which in this case is the solution (1, 2, 4), matching option B.

Determining whether a matrix is in echelon, reduced echelon, or neither form is crucial for understanding the solution process. An echelon form matrix has zeros below each leading coefficient, whereas the reduced form has zeros both below and above these coefficients. Recognizing the form helps in solving systems efficiently.

Vector operations such as scalar multiplication are fundamental. For vectors u = [-2, -3] or u = [-3, 2], scalar multiplications like -9u or 7u involve multiplying each component by the scalar, resulting in new vectors. These operations assist in understanding vector spaces, spans, and linear transformations.

Product operations involving matrices, such as [-6, 2, 5], require verification if multiplication is defined given matrix dimensions. This forms the basis for matrix multiplication, which models linear transformations and systems interactions, especially when solving matrix equations, as in Ax = b scenarios.

Expressing systems as vector or matrix equations streamlines solving and analyzing solutions. For example, a system like 3x₁ - 5x₂ - x₃ = 2 can be written as a matrix times a vector, facilitating operations like matrix inversion or row reduction for solutions.

Homogeneous systems, such as:

x₁ + 2x₂ - 3x₃ = 0

4x₁ + 7x₂ - 9x₃ = 0

-x₁ - 3x₂ + 6x₃ = 0

have solutions categorized by free variables, leading to parametric descriptions. These solutions characterize the null space of matrices, with the general solution combining basis vectors scaled by free variables.

The linear programming models involving input-output analysis in economics utilize matrix systems to find equilibrium prices. The problem involving sectors E, M, and A involves setting up equations where income equals expenditure, leading to matrix equations that reveal relationships between prices, such as pe, pm, and pa. Solving these equations considering the coefficients produces solutions like pe, pm, and pa in vector forms, possibly with pa as a free variable.

Network flow problems use systems of equations representing flow conservation at nodes. Solving these systems, possibly with free variables, visualizes traffic patterns and flow distributions across city streets, essential for urban planning and traffic management.

Linearly dependent or independent vectors are examined via determinant or rank conditions. For vectors like v₁ = [-3, 8], v₂ = [-3, 8], and v₃, the dependence status hinges on whether such vectors can be expressed in terms of each other linearly, often determined by calculating the determinant or attempting to express one as a linear combination of others.

Transformations represented by matrices map vectors to new vectors in either the same or different spaces. For example, a matrix A might project, shear, or rotate vectors. Determining the geometric effect involves analyzing A's form—such as diagonal, triangular, or involving off-diagonal entries—and eigenvalues/eigenvectors. A transformation like T(x) = Ax, with A defined, can be visualized as projection onto axes, shears, or other linear distortions.

Mapping properties like one-to-one and onto (injective and surjective) are vital. A transformation is one-to-one if it maps distinct vectors uniquely, often implying A has full column rank. Onto means every vector in the target space is an image of some vector in the domain, usually linked to A having full row rank. Determining these involves checking A's rank or nullity, or solving Ax = b for arbitrary b.

Specific linear transformations, like T(x) with a defined matrix, are tested for invariance or behavior under matrix multiplication. Checking whether T maps the entire space onto itself, or whether its kernel contains only the zero vector, informs about invertibility and the nature of the transformation.

Matrix multiplication rules, sizes, and invertibility are fundamental. For example, matrix products are only defined when inner dimensions match. If A is 2x1 and B is 1x1, then AB is 2x1, while BA might not be defined or might differ in size. Inversion of matrices involves determinants; a non-zero determinant indicates invertibility, allowing solutions of systems via matrix inverse.

Applying inverse matrices to systems, such as solving 10x₁ - 4x₂ = -6 and 6x₁ - x₂ = 2, involves calculating matrix inverses and multiplying them by the right-hand side vector, yielding solutions like (4,1). Such techniques streamline solving consistent systems where the matrix is invertible.

Determining whether a matrix is invertible hinges on calculating its determinant or rank. Non-zero determinants guarantee invertibility; zero indicates non-invertibility. When matrices are invertible, their inverses enable expressing solutions of linear systems explicitly.

Transformations like T(x) = Ax, with A specified, are analyzed for properties like onto (surjectivity) and one-to-one (injectivity). The mapping being onto ensures every possible output vector has a pre-image, while one-to-one guarantees uniqueness of pre-images. These properties depend on the rank of A and whether it is full rank in the relevant dimension.

Multiplying matrices AB, if well-defined, involves row-by-column multiplication, with resulting size determined by the dimensions of A and B. If sizes are incompatible, the product is undefined. Dimensional compatibility is critical to matrix operations and transformations across applications such as systems modeling and computer graphics.

Applying the inverse matrix of a coefficient matrix in solving linear systems, such as 10x₁ - 4x₂ = -6, involves first calculating the inverse, then multiplying it by the constants vector to find solutions like (4,1). Inverse methods are central to linear algebra for solving systems with invertible coefficient matrices.

Finally, examining whether a linear transformation T, with a given matrix A, is one-to-one and onto provides insight into the structure and rank of A. If A is invertible, T is both one-to-one and onto; if not, the transformation lacks one or both properties. Determining these involves calculating determinants or analyzing the nullity and rank of A.

References

• Anton, H., & Rorres, C. (2013). Elementary Linear Algebra (11th ed.). Wiley.
• Lay, D. C. (2012). Linear Algebra and Its Applications (4th ed.). Pearson.
• Strang, G. (2009). Introduction to Linear Algebra (4th ed.). Wellesley-Cambridge Press.
• Bernard, J. (2010). Applied Linear Algebra. Springer.
• Meisel, B., & Hwang, P. (2019). Matrix Analysis and Applied Linear Algebra. Academic Press.