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Determine which of the following are linear transformations from R4 to R³. For any which are not, give a reason why not. For any which are, find the kernel (also known as null space) and image, and verify that the dimension of the kernel plus the dimension of the image is 4: 3x ~)-()~() x +2y + z (ii) 0₂ 2x+yz - 2t (i) 0₁ X t = ()-( t (iii) 03 N X t xy xz + 2yt 3xt + 3z X- - 2z+t 2+y+z+t 2y +7z

Determine which of the following are linear transformations from R4 to R³. For any which are not, give a reason why not. For any which are, find the kernel (also known as null space) and image, and verify that the dimension of the kernel plus the dimension of the image is 4: 3x ~)-()~() x +2y + z (ii) 0₂ 2x+yz - 2t (i) 0₁ X t = ()-( t (iii) 03 N X t xy xz + 2yt 3xt + 3z X- - 2z+t 2+y+z+t 2y +7z

Advanced Engineering Mathematics
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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Determine which of the following are linear transformations from R4 to R³. For any which are not, give a reason why not. For any which are, find the kernel (also known as null space) and image, and verify that the dimension of the kernel plus the dimension of the image is 4: X 3x ~()- () -() 0₁ x + 2y + z (ii) 0₂ 2x+y-z - 2t t t ()-( t (iii) 03 xy xz + 2yt 3xt + 3z - 2z+t 2+y+z+t 2y +7z X-
Transcribed Image Text:Determine which of the following are linear transformations from R4 to R³. For any which are not, give a reason why not. For any which are, find the kernel (also known as null space) and image, and verify that the dimension of the kernel plus the dimension of the image is 4: X 3x ~()- () -() 0₁ x + 2y + z (ii) 0₂ 2x+y-z - 2t t t ()-( t (iii) 03 xy xz + 2yt 3xt + 3z - 2z+t 2+y+z+t 2y +7z X-
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