Mark the following either TRUE or FALSE. Justification is required if FALSE.(a) If A is a 4 x 7 matrix and T is defined by T(x) = Ax, then the range ofT is a subset of R¹.(b) Every matrix transformation is a linear transformation.(c) Every linear transformation is a matrix transformation.

Answered: Image /qna-images/answer/41d91c62-202e-4ecf-8ea4-3ef21ef08c62.jpg

Answered: Mark the following either TRUE or…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let T be the linear transformation whose standard matrix is given. Decide if T is a one-to-one mapping. Justify your answer. Choose the correct answer below. -7 6 9 5 2 11 -5 2 -6 - 3 - 1 10 3-15 9-25 O A. The transformation T is one-to-one because the equation T(x) = 0 has only the trivial solution. B. The transformation T is one-to-one because the equation T(x) = 0 has a nontrivial solution. O C. The transformation T is not one-to-one because the equation T(x) = 0 has a nontrivial solution. O D. The transformation T is not one-to-one because the equation T(x) = 0 has only the trivial solution.
Describe in a few sentences a method to determine the standard matrix A for this transformation. Recall that is the matrix A such that T(v) = Av.
Determine whether the statement below is true or false. Justify the answer. If A is an mxn matrix, then the range of the transformation x→Ax is Rm. Choose the correct answer below. O A. The statement is true. The range of the transformation is RM, because each vector in RM is a linear combination of the columns of A. OB. The statement is false. The range of the transformation is R" because the domain of the transformation is RM OC. The statement is true. The range of the transformation is RM, because each vector in RM is a linear combination of the rows of A. O D. The statement is false. The range of the transformation is the set of all linear combinations of the columns of A, because each image of the transformation is of the form Ax.
Give an example of each of the following, or explain why no such example exists. (If you give an example, you do not need to provide any additional explanation; just give the example.) 1 1 1 (a) A matrix whose inverse is 2 2 2 3 3 3 (b) The matrix for an onto linear transformation from R¹ to R³. (c) The matrix for a one-to-one linear transformation from R¹ to R³.
7
Let T be the linear transformation represented by the matrix (The matrix is shown in the image). Find the kernel and image of this transformation. Does the dimension theorem hold? It is this invertible linear transformation, if so find the inverse linear transformation. Note: In the image the exercise is clearer, do not skip any step to arrive at the result.
Consider a matrix A with m rows. Which condition below will ensure that every vector in Rm is in the span of the columns of A? The linear transformation with standard matrix A is injective. A has no zero row. Every row of A is a pivot row. Every column of A has a pivot.

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Mark the following either TRUE or FALSE. Justification is required if FALSE. (a) If A is a 4 x 7 matrix and T is defined by T(x) = Ax, then the range of T is a subset of R¹. (b) Every matrix transformation is a linear transformation. (c) Every linear transformation is a matrix transformation.