Worksheet: Image and Kernel of Linear Transformation

In this worksheet, we will practice finding the image and basis of the kernel of a linear transformation.


Suppose that the columns of an 𝑚×𝑚 matrix 𝐴 are linearly independent. Then, which of the following statements is always true?

  • AThe determinant of 𝐴 is 1.
  • B𝐴 is a projection matrix.
  • C𝐴 has 𝑚 linearly independent eigenvectors.
  • DThe kernal of 𝐴 is {}0.
  • E𝐴 is idempotent.


Find a basis for ker(𝐴) , where 𝐴=12321021121443302112.

  • A122100,121010,11001
  • B100212,010112,00111
  • C10011,01011,00111
  • D212100,112010,11001
  • E11100,11010,11001


Fill in the blank. A linear map 𝑇𝐿(𝑉,𝑊) is , if and only if 𝑇 is injective and surjective.

  • Ainvertible
  • Bone-to-one
  • Ccontinuous
  • Dsingular


Let 𝑇𝐿 be such that 𝑇𝑥𝑦=𝑦𝑥𝑥𝑦.forall Then, 𝑇 is .

  • Aa surjective linear map
  • Ba zero map
  • Ca vector space
  • Dan identity map

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