Solved: In Exercises 14, let T(x) = Ax for the given matrix A, and findT(u1) and T(u2)

In Exercises 14, let T(x) = Ax for the given matrix A, and find T(u1) and T(u2) for the given u1 and u2. A =  2 1 3 5 , u1 =  4 2  , u2 =  1 6 

In Exercises 14, let T(x) = Ax for the given matrix A, and find T(u1) and T(u2) for the given u1 and u2. A = 1 0 2 4 3 3 , u1 =  1 2  , u2 =  5 0 

In Exercises 14, let T(x) = Ax for the given matrix A, and find T(u1) and T(u2) for the given u1 and u2. A =  0 4 2 3 1 2  , u1 = 3 2 1  , u2 =  4 5 2

In Exercises 14, let T(x) = Ax for the given matrix A, and find T(u1) and T(u2) for the given u1 and u2. A = 2 5 2 0 1 2 0 1 1  , u1 =  0 7 2  , u2 =  3 5 1 

In Exercises 58, determine if the given vector is in the range of T(x) = Ax, where A =  1 2 0 3 21 y =  3 6 

In Exercises 58, determine if the given vector is in the range of T(x) = Ax, where A =  1 2 0 3 21 y =  1 4 

In Exercises 58, determine if the given vector is in the range of T(x) = Ax, where A =  1 2 0 3 21 y =  2 7 

In Exercises 58, determine if the given vector is in the range of T(x) = Ax, where A =  1 2 0 3 21 y =  4 5

Suppose that a linear transformation T satisfies T(u1) =  2 1  , T(u2) =  3 2  Find T(2u1 + 3u2).

Suppose that a linear transformation T satisfies T(u1) =  3 1 2  , T(u2) = 1 1 4  Find T(3u1 2u2).

Suppose that a linear transformation T satisfies T(u1) =  3 0  , T(u2) =  2 1  , T(u3) =  0 5  Find T(u1 + 4u2 3u3).

Suppose that a linear transformation T satisfies T(u1) =  3 1 2  , T(u2) = 1 1 4  , T(u3) = 6 0 0  Find T(u1 + 4u2 2u3).

In Exercises 1320, determine if the given function is a linear transformation. If so, identify the matrix A such that T(x) = Ax. If not, explain why not. T(x1, x2) = (3x1 + x2, 2x1 + 4x2)

In Exercises 1320, determine if the given function is a linear transformation. If so, identify the matrix A such that T(x) = Ax. If not, explain why not. T(x1, x2) = (x1 x2, x1x2)

In Exercises 1320, determine if the given function is a linear transformation. If so, identify the matrix A such that T(x) = Ax. If not, explain why not. T(x1, x2, x3) = (2 cos(x2), 3 sin(x3), x1)

In Exercises 1320, determine if the given function is a linear transformation. If so, identify the matrix A such that T(x) = Ax. If not, explain why not. T(x1, x2, x3) = (5x2, 7x3)

In Exercises 1320, determine if the given function is a linear transformation. If so, identify the matrix A such that T(x) = Ax. If not, explain why not. T(x1, x2, x3) = (4x1 + x3, 6x1 + 5x2)

In Exercises 1320, determine if the given function is a linear transformation. If so, identify the matrix A such that T(x) = Ax. If not, explain why not. T(x1, x2, x3) = (x1 + 3x2 + x3, 2x1 + 7x2 + 4, 3x3)

In Exercises 1320, determine if the given function is a linear transformation. If so, identify the matrix A such that T(x) = Ax. If not, explain why not. T(x1, x2) = (x2 sin(/4), x1 ln(2))

In Exercises 1320, determine if the given function is a linear transformation. If so, identify the matrix A such that T(x) = Ax. If not, explain why not. T(x1, x2) = (3x2, x1 + 5|x2|, 2x1)

For Exercises 2128, let T(x) = Ax for the given matrix A. Determine if T is one-to-one and if T is onto. A =  1 3 2 5

For Exercises 2128, let T(x) = Ax for the given matrix A. Determine if T is one-to-one and if T is onto. A =  3 2 9 6

For Exercises 2128, let T(x) = Ax for the given matrix A. Determine if T is one-to-one and if T is onto. A =  5 4 2 3 1 0

For Exercises 2128, let T(x) = Ax for the given matrix A. Determine if T is one-to-one and if T is onto. A =  1 32 4 12 8 

For Exercises 2128, let T(x) = Ax for the given matrix A. Determine if T is one-to-one and if T is onto. A =  1 2 3 5 2 7 

For Exercises 2128, let T(x) = Ax for the given matrix A. Determine if T is one-to-one and if T is onto. A =  2 4 5 10 4 8

For Exercises 2128, let T(x) = Ax for the given matrix A. Determine if T is one-to-one and if T is onto. A = 2 84 3 23 1 14 5

For Exercises 2128, let T(x) = Ax for the given matrix A. Determine if T is one-to-one and if T is onto. A =  1 2 5 3 7 8 2 4 6

For Exercises 2932, suppose that T(x) = Ax for the given A. Sketch a graph of the image under T of the unit square in the first quadrant of R2. A =  3 0 0 3

For Exercises 2932, suppose that T(x) = Ax for the given A. Sketch a graph of the image under T of the unit square in the first quadrant of R2. A =  2 0 0 4

For Exercises 2932, suppose that T(x) = Ax for the given A. Sketch a graph of the image under T of the unit square in the first quadrant of R2. A =  1 2 3 1

For Exercises 2932, suppose that T(x) = Ax for the given A. Sketch a graph of the image under T of the unit square in the first quadrant of R2. A =  3 1 6 2 

FIND AN EXAMPLE For Exercises 3338, find an example that meets the given specifications. A linear transformation T : R2 R2 such that T 1 0  =  2 3 

FIND AN EXAMPLE For Exercises 3338, find an example that meets the given specifications. A linear transformation T : R2 R3 such that T 0 1  = 1 4 5 

FIND AN EXAMPLE For Exercises 3338, find an example that meets the given specifications. A linear transformation T : R2 R2 such that T 3 1  =  7 0 

FIND AN EXAMPLE For Exercises 3338, find an example that meets the given specifications. A linear transformation T : R3 R2 such that T  3 1 2  =  7 1 

FIND AN EXAMPLE For Exercises 3338, find an example that meets the given specifications. A linear transformation T : R2 R2 such that T 2 1  =  0 7  and T 1 3  =  5 6 

FIND AN EXAMPLE For Exercises 3338, find an example that meets the given specifications. A linear transformation T : R2 R2 such that T 1 2  =  1 8  and T  2 3  =  2 13

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. The codomain of a linear transformation is a subset of the range.

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. The range of a linear transformation must be a subset of the domain.

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. If T is a linear transformation and v is in range(T), then there is at least one u in the domain such that T(u) = v.

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. If T(x) is not a linear transformation, then T(r x) = r T(x) for all r and x.

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. The function T(x) = Ax + b is a linear transformation only when b = 0.

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. . If T : R2 R2 is a linear transformation, then the image under T of the unit square in the first quadrant will be a parallelogram.

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. If T1(x) and T2(x) are one-to-one linear transformations from Rn to Rm, then so is W(x) = T1(x) + T2(x).

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. If T1(x) and T2(x) are onto linear transformations from Rn to Rm, then so is W(x) = T1(x) + T2(x).

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. If a linear transformation T : R4 R4 is one-to-one, then T(x) = 0 has nontrivial solutions.

TRUE OR FALSE For Exercises 3948, determine if the statement is true or false, and justify your answer. If a linear transformation T : R3 R3 is one-to-one, then T also must be onto.

A linear transformation T : R2 R2 is called a dilation if T(x) = r x for r > 1. (It is called a contraction if 0 < r < 1.) (a) Find the matrix A such that T(x) = Ax. (b) Letr = 2, and then sketch the graphs of x =  2 1  and T(x).

Suppose that T : R3 R2 is given by T x1 x2 x3  =  x1 x2  The T is called a projection transformation because it projects vectors in R3 onto R2. (a) Prove that T is a linear transformation. (b) Find the matrix A such that T(x) = Ax. (c) Describe the set of vectors in R3 such that T(x) = 0.

. Suppose that x = (x1, ... , xn) and y = (y1, ... , yn) are vectors in Rn. Then the dot product of x and y is given by x y = x1 y1 ++ xn yn Now let u be a fixed vector in Rn, and define T(x) = u x. Show that T is a linear transformation.

Suppose that x = (x1, x2, x3) and y = (y1, y2, y3) are vectors in R3. Then the cross product of x and y is given by x y = x2 y3 x3 y2 x3 y1 x1 y3 x1 y2 x2 y1  Now let u be a fixed vector in Rn, and define T(x) = u x. Show that T is a linear transformation.

Suppose that T : R3 R2 is a linear transformation. Prove that T is not one-to-one.

Suppose that T : R2 R4 is a linear transformation. Prove that T is not onto.

Suppose that T is a linear transformation. Show that T(0) = 0.

Suppose that T(x) = Ax is a linear transformation and that there exists u = 0 such that T(u) = 0. Show that the columns of A must be linearly dependent.

Suppose that T(x) = Ax. Show that T(ru) = r T(u) for all scalars r and all vectors u.

Suppose that T : Rm Rn. (a) Show that if T is a linear transformation, then T(r x + s y) = r T(x) + s T(y) for all scalars r and s and all vectors x and y. (b) Now show the converse: If T(r x + s y) = r T(x) + s T(y) or all scalars r and s and all vectors x and y, then T is a linear transformation.

Prove that a linear transformation T is one-to-one if and only if T(u) = T(v) implies that u = v.

Suppose that T is a linear transformation and that u1 and u2 are linearly dependent. Prove that T(u1) and T(u2) are also linearly dependent.

Suppose that T is a linear transformation and that T(u1) and T(u2) are linearly independent. Prove that u1 and u2 must be linearly independent.

Suppose that T is a linear transformation and that u1 and u2 are linearly independent. Show that T(u1) and T(u2) need not be linearly independent.

Suppose that y is in the range of a linear transformation T and that there exists a vector u = 0 such that T(u) = 0. Show that there are infinitely many solutions x to T(x) = y. (HINT: First, explain why T(ru) = 0 for any scalar r, and then show that T(x + ru) = y when T(x) = y.)

. Let u and v be two distinct vectors in R2. Show that the set of points on the line segment connecting u and v is the same as the set of points (1 s)u + s v, 0 s 1

Let T : R2 R2 be a linear transformation with T(x) = Ax. Prove that the image of the unit square in the first quadrant is a parallelogram if the columns of A are linearly independent, and a line segment (possibly of zero length) if the columns of A are linearly dependent.

In graph theory, an adjacency matrix A has an entry of 1 at ai j if there is an edge connecting node i with node j, and a zero otherwise. (Such matrices come up in network analysis.) Suppose that a graph with five nodes has adjacency matrix A = 01011 10101 01010 10100 11000 Let T : R5 R5 be given by T(x) = Ax. (a) Describe how to use T(x) to determine the number of edges connected to node j. (b) How can one use T(x) to help determine the total number of graph edges? 67. (Calculus required) Suppose that for each polynomial of degree 2 or less, we identify the coefficients with a vector in R3 by ax2 + bx + c a b c  (a) Show that addition of polynomials corresponds to vector addition and that multiplication of a polynomial by a constant corresponds to scalar multiplication of a vector. (b) Let T : R3 R3 be the function that takes a polynomial vector as input and produces the vector of the derivative as output. Prove that T is a linear transformation. (c) Find the matrix A such that T(x) = Ax. (d) Is T one-to-one? Onto? Give a proof or counter-example for each.

(Calculus required) Suppose that for each polynomial of degree 2 or less, we identify the coefficients with a vector in R3 by ax2 + bx + c a b c  (a) Show that addition of polynomials corresponds to vector addition and that multiplication of a polynomial by a constant corresponds to scalar multiplication of a vector. (b) Let T : R3 R3 be the function that takes a polynomial vector as input and produces the vector of the derivative as output. Prove that T is a linear transformation. (c) Find the matrix A such that T(x) = Ax. (d) Is T one-to-one? Onto? Give a proof or counter-example for each.

(Calculus required) Complete Exercise 67 for polynomials of degree 3 or less identified with vectors in R4. (Calculus required) In Chapter 9 we extend the concept of a linear transformation by observing that the two conditions given in Definition 3.1 exist for other types of mathematical operations. Exercises 6970 provide a sneak preview. Assume that f (x) and g (x) are in the set of functions C(R) that have infinitely many derivatives on R and that r is a real number.

Let T : C(R) C(R) be defined by T  f (x)  = f (x). (a) Evaluate T  x2 + sin(x)  . (b) Prove that T satisfies conditions analogous to those given in Definition 3.1: i. T  f (x) + g (x)  = T  f (x)  + T  g (x)  ii. T  r f (x)  = r T f (x) 

Let T : C(R) R be defined by T  f (x)  =  1 0 f (x) dx (a) Evaluate T  4x3 6x2 + 1  . (b) Prove that T satisfies conditions analogous to those given in Definition 3.1: i. T  f (x) + g (x)  = T  f (x)  + T  g (x)  ii. T  r f (x)  = r T f (x) 

C In Exercises 7174, refer to the MP3 scenario given at the beginning of the section. Use the linear transformation T to determine the cost vector that results from producing the specified number of J8s, J40s, and J80s. 5 J8s, 3 J40s, and 6 J80s.

C In Exercises 7174, refer to the MP3 scenario given at the beginning of the section. Use the linear transformation T to determine the cost vector that results from producing the specified number of J8s, J40s, and J80s. 6 J8s, 4 J40s, and 10 J80s.

C In Exercises 7174, refer to the MP3 scenario given at the beginning of the section. Use the linear transformation T to determine the cost vector that results from producing the specified number of J8s, J40s, and J80s. 14 J8s, 10 J40s, and 9 J80s.

C In Exercises 7174, refer to the MP3 scenario given at the beginning of the section. Use the linear transformation T to determine the cost vector that results from producing the specified number of J8s, J40s, and J80s. 16 J8s, 20 J40s, and 18 J80s.

C For Exercises 7580, let T(x) = Ax for the given matrix A. Determine if T is one-to-one and if T is onto. A = 4 2 526 7 2 0 4 1 0 3 5 7 1

C For Exercises 7580, let T(x) = Ax for the given matrix A. Determine if T is one-to-one and if T is onto. A =  4 2 5 21 5 14 4 5 8 1 6 2 3 2

C For Exercises 7580, let T(x) = Ax for the given matrix A. Determine if T is one-to-one and if T is onto. A = 2 140 3 311 1 183 0 214

C For Exercises 7580, let T(x) = Ax for the given matrix A. Determine if T is one-to-one and if T is onto. A = 3 20 5 0 12 3 2 13 1 4 2 3 1

C For Exercises 7580, let T(x) = Ax for the given matrix A. Determine if T is one-to-one and if T is onto. A = 2 35 1 6 03 2 4 21 1 8 23 4 1 25 3

C For Exercises 7580, let T(x) = Ax for the given matrix A. Determine if T is one-to-one and if T is onto. A = 4 3 2 9 101 1 3 0 2 4 2 433 5 703