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.

Q1:

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.
  • B 𝐴 is a projection matrix.
  • C 𝐴 has 𝑚 linearly independent eigenvectors.
  • DThe kernal of 𝐴 is {}0.
  • E 𝐴 is idempotent.

Q2:

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

  • A 1 2 2 1 0 0 , 1 2 1 0 1 0 , 1 1 0 0 1
  • B 1 0 0 2 1 2 , 0 1 0 1 1 2 , 0 0 1 1 1
  • C 1 0 0 1 1 , 0 1 0 1 1 , 0 0 1 1 1
  • D 2 1 2 1 0 0 , 1 1 2 0 1 0 , 1 1 0 0 1
  • E 1 1 1 0 0 , 1 1 0 1 0 , 1 1 0 0 1

Q3:

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

Q4:

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

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

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.