L A+Assignment 3

National University Of Computer & Emerging Sciences-Faisalabad Chiniot

Assignment 3

Linear Algebra

Submission date: 27-10-2015

______________________________________________________________________________ Q#1 Let

be a transformation. Find the standard matrix a. b. c. d.

that gives us

Reflection through the line . Reflection through the origin. Projection onto the If first reflects points through the vertical and then rotates points through an angle of ⁄ radians about origin in counter clockwise direction.

first rotates points through an angle of ⁄ radians about origin in counter clockwise direction and then reflects points through the vertical . f. If rotates points (2, 3) & (2,  3) through ⁄ radians about origin in counter clockwise direction. g. If first rotates points (2, 3) & (2,  3) through ⁄ radians about origin in counter clockwise direction and then reflects points through the horizontal . e. If

______________________________________________________________________________ Q#2 Let

be a transformation such that a. b. c. d.

([ ])

[

]

Determine either the given transformation is linear or not. Find matrix of transformation. Is T one-to one? Is T maps onto

e. Find a vector

whose image is [

]

f. Find the image of [ ] g. Determine either the columns of are linearly dependent or independent. Page 1 of 2

h. Is co-domain=range? explain in detail. i. Find a vector that belongs to the codomain but does not belongs to the range. ______________________________________________________________________ Q#3 If

[

]

a. Give geometric description of the transformation x  Ax . b. Find the order of the transformation. c. Let A be a matrix that with the property that linear transformation ONTO . Explain why the transformation must be one to one.

maps

______________________________________________________________________________ Q#4 Find all x  R 4 that are mapped into the zero vector by the transformation

x  Ax for the matrix

3 1 A  0  1

2 10  6 0 2  4 . 1 2 3  4 10 8 

Is U in the subset of R 3 spanned by the columns of A ? ______________________________________________________________________________ Q#5

 1 0 Let b    and let A be the matrix given above. Is b in the range of linear 0   4 transformation x  Ax ? Why or why not?

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