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Linear Algebra With Applications
Found in: Page 54
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Give a geometric interpretation of the linear transformations defined by the matrices in Exercises 16 through 23. Show the effect of these transformations on the letter L considered in Example 5. In each case, decide whether the transformation is invertible. Find the inverse if it exists, and interpret it geometrically. See Exercise 13.

20. [0110]

The matrix [0110] has scalable change in shape of L and is invertible with inverse .


The geometrical interpretation is:

See the step by step solution

Step by Step Solution

Step by Step Explanation: Step 1: Consider the matrix.

Let the matrix be,


The letter L is made up of vectors 10 and 02

Step 2: Compute the vectors.

Let the matrix be,


Consider the vector 10

role="math" localid="1659689477902" T(x)=[0110]xT[10]=[0110][10]T[10]=[01]

Consider the vector 02


Step 3: Graph the letter using matrix.

Now, graph the original vectors and the obtained vectors as follow:

T(x) is obtained by scaling the vector x.

Step 4: Check for the invertibility of the matrix and find the inverse if exists.

The matrix [abcd] is invertible if and only if ad-bc0 .

The inverse of the matrix [abcd] is role="math" localid="1659690085262" [abcd]-1=1ad-bc[d-b-ca] .

Consider the matrix,


Therefore, the matrix is invertible.

The inverse of the matrix is,


Therefore, the matrix [0110] is invertible and its inverse is [0110], and the shape of L gets transformed in terms of scalability

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