Question

(a) (1 point) State the definition of a linear transformation. (Hint: this should involve two important properties of the transformation....) (b) (2 points) Is the map T : R^3 -> R^2 given by T(x, y, z) = (x + y, 2^z - 1) a linear transformation? Prove your answer using the definition from part a). (c) (3 points) Verify using the definition that the map T : R^3 -> R^2 given by T(x, y, z) = (x + y + z, x) is a linear transformation. (d) (2 points) Write down the matrix for the linear transformation from part (c), showing your work. (e) (4 points) Explain carefully why that if a matrix is not square, then the corresponding linear transformation can be either injective or surjective, but it cannot be both.

          (a) (1 point) State the definition of a linear transformation. (Hint: this should involve two important properties of the transformation....)
(b) (2 points) Is the map T : R^3 -> R^2 given by
T(x, y, z) = (x + y, 2^z - 1)
a linear transformation? Prove your answer using the definition from part a).
(c) (3 points) Verify using the definition that the map T : R^3 -> R^2 given by
T(x, y, z) = (x + y + z, x)
is a linear transformation.
(d) (2 points) Write down the matrix for the linear transformation from part (c), showing your work.
(e) (4 points) Explain carefully why that if a matrix is not square, then the corresponding linear transformation can be either injective or surjective, but it cannot be both.

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(a) (1 point) State the definition of a linear transformation. (Hint: this should involve two important properties of the transformation....)
(b) (2 points) Is the map T : R^3 -> R^2 given by
T(x, y, z) = (x + y, 2^z - 1)
a linear transformation? Prove your answer using the definition from part a).
(c) (3 points) Verify using the definition that the map T : R^3 -> R^2 given by
T(x, y, z) = (x + y + z, x)
is a linear transformation.
(d) (2 points) Write down the matrix for the linear transformation from part (c), showing your work.
(e) (4 points) Explain carefully why that if a matrix is not square, then the corresponding linear transformation can be either injective or surjective, but it cannot be both.

Added by Jennifer S.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Hoan Nguyen

03/08/2023

Step 1

a) A transformation T: V → W is called a linear transformation if it satisfies the following two properties for all vectors u, v ∈ V and scalar c: Show more…

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(a) State the definition of a linear transformation. (Hint: this should involve two important properties of the transformation....) (b) Is the map T : R^3 -> R^2 given by T(x, y, z) = (x + y, 2^z - 1) a linear transformation? Prove your answer using the definition from part a). (c) Verify using the definition that the map T : R^3 -> R^2 given by T(x, y, z) = (x + y + z, x) is a linear transformation. (d) Write down the matrix for the linear transformation from part (c), showing your work. (e) Explain carefully why that if a matrix is not square, then the corresponding linear transformation can be either injective or surjective, but it cannot be both.

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00:04 All right, so in part a, the definition of a linear transformation.

00:10 So we have the following definition, the map t from vector space v to a vector space w is a linear transformation.

00:26 If it satisfy the following two properties.

00:31 One, t of u plus v is t of u plus t of v for all u and v.

00:40 Vector in capital v vector space.

00:44 And the second property is that t of a u is a time t u for u in v and a in the real number which is the scalar.

00:59 So part b.

01:01 So we have the map t of x y z is given.

01:05 So this is the solution.

01:08 It's given by x plus y to to z minus 1.

01:18 So now we have tf 0 0 0 is given by 0 plus 0, 2 to 0 minus 1.

01:36 Okay, we shouldn't consider t of this vector.

01:39 So tf 1, 1, 1 is given by 1 plus 1, 2 to 1 minus 1, which is a vector 2 1.

01:51 So now t f 2 to 2 we plug in 2 plus 2 2 to 2 to 2 to the 2 minus 1 which is a vector 4 3 so now 2 time t f1 1 1 1 1 is 2 time 2 1 which is the vector 4 2 that different from the vector 4 3 which is t 2 2 2 2 2 2 2 2 2.

02:22 So t does not satisfy property to this property, therefore t is not a linear transformation but c where t of x, y, z is equal to x plus y plus c, x.

03:03 So we need to show that this is a linear transformation...

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